Автор: Нагель Э. Ньюмен Дж. Р.
Теги: основы математики математическая логика математика
ISBN: 978-5-396-00092-6
Год: 2010
Текст
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6e3 OTBeTa. B qHCJIe aKCHOM, H,a 6a3e KOTOpbIX CTpOH-
JIaCb eBKJIHAOBa CHCTeMaTH3an:IDI reOMeTpHH, HMeeTCH
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iKeHHOH EBKJIH,nOM cpOpM)'JIHpOBKe 3Ta aKCHOMa paB-
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u u
tITO qepe3 TOq JIeiKa BHe ,naHHOH npHMOH, MOiKHO
npOBeCTH e,nHHCTBeHHYIO npHMyK>, napaJIJIeJIbHYlO ,l:\aH-
HOH npHMOH. EIne aHTHqHbIM MaTeMaTHKaM 3Ta aKCHOMa
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CTBHTeJIbHO nOJIyqHTb HCKOMoe ,nOKa3aTeJIbCTBO AJUI aK-
14
2. np06AeMa HenpoTHBope HBocTH
CHOMbI napaJIJIeJIbHoCTH? IloKoJIeHIDI MaTeMaTHKOB 6e3-
ycnemHo nbITaJIHCb OTBeTHTb Ha 3TOT Bonpoc. Ho Heo -
HOKpaTHbIe Hey aqH nonbITOK nocTpoeHIDI HCKOMoro o-
Ka3aTeJIbCTBa He 03HaqaJIH e e, qTO HHKTO He npeycneeT
B 3TOM eJIe 60JIbme, qeM B BIDKHOH AJUI qeJIOBeqeCTBa
np06JIeMe H306peTeHIDI 6e30TKa3HO H Ha Bce BpeMeHa
AeHCTByro ero cpe CTBa OT HaCMopKa. TaKoe nOJlO:>Ke-
HHe Be eH npOAOJI:>KaJIOCb O cepeAHHbI XIX CTOJIeTIDI -
O Tex nop, nOKa B pa60Tax raycca, EOHaH, JI06aqeB-
cKoro, PHMaHa H pyrHX MaTeMaTHKOB He 6bLna OKa-
3aHa Hea03MO:NCHOCmb BbIBOAa aKCHOMbI napaJIJIeJIbHO-
CTH H3 OCTaJIbHbIX aKCHOM eBKJIHAOBOH reOMeTpHH. 9TOT
pe3YJIbTaT HMeJI rpOM(lAHOe 3HaqeHHe AJUI nOHHMaHIDI
npHpOAbI Hamero MbIllIJIeHIDI. B nepByro Oqepe b OH
npHBJIeK BHHMaHHe K TOM)' nOpa3HTeJIbHOMY <paKTY, qTO
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B03MO:>KHOCTH AOKa3aTeJIbCTBa HeKOTopbIX apH<pMeTHqe-
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ro, pa3pemeHHe CTapOH np06JIeMbI 06 aKCHOMe napaJI-
JIeJIbHOCTH HeH36e:>KHO npHBOAHJIO K BbIBOA)', qTO aK-
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napaJIJIeJIbHbIX on eHHeM, COrJIaCHO KOTOpOM)' qepe3
15
2. np06J\eMa HenpoTHBope4HBocTH
v v
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K7IH lKe, HanpoTHB, ,non eHHeM, COrJIaCHO KOTOPOMY
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cOBepllIeHHo nO,l:\OpBaHo. EOJIee Toro, nOCTeneHHO CTaJIO
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v
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KO cnoc06Ha BBO,l:\RTb B 3a6JI eHHe H HHKOHM 06pa30M
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16
2. np06AeMa HenpoTwBope4HBocTH
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nOCKOJIbK)' MaTeMaTHt.IeCKHe npe,n:.nO:>KeHIDI B npHH H-
ne MOryr 6bITb HCTOJIKOBaHbI CKopee KaK yrBep eHIDI
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17
2. np06AeMa HenpoTHBope4HBocTH
/102U1.LeCKU Heo6xoOUM.blMU C/leOCmBWl.M.U H3 HaqaJIbHbIX
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. MO)KHO CKa3aTb, 4TO nepBOHa4aAbHbie TepMHHbl «HeSlBHO» onpeAeAeHbl
aKCHOMaMH H 4TO Bce, 4TO He nOKpblBaeTCSI 3TUMU HeSlBHblMH onpeAeAe-
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18
2. np06AeMa HenpoTHBope4HBocTH
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19
2. np06/\eMa HenpOTJ.1BOpe4J.1BOCTJ.1
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20
2. np06AeMa HenpOTJIIBOpe4J11BOCTJII
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CTpaKTHbIX nocryJIaTOB. PaCCMOTpHM, HanpHMep, CJIe,ny-
IO CHCTeMY nocryJIaTOB, B <POPMYJIHpOBKH KOTOpbIX
BXOAHT ,ZJ;Ba KJIacca K H L, n01VIHHHaH «npHpo,n;a» KOTO-
u
pbIX OCTaeTCH Heonpe,n;eJIeHHOH, eCJIH He CqHTaTb Toro,
qTO caMH nocryJIaTbI «HeHBHO» onpe,n;eJIHlOT 3TH KJIaCCbI.
1. JII06bIe ,ZJ;Ba (pa3JIHqHbIX) tL1IeHa KJIacca K npHHa,n;-
JIe)KaT B TOqHOCTH O,ZJ;HOM)' tL1IeH)' KJIacca L.
2. HH O,ZJ;HH tL1IeH KJIacca K He npHH3AJIe)KHT 60JIee
qeM ,ZJ;ByM (pa3JIHqHbIM) tL1IeHaM KJIacca L.
3. He Bce ttJleHbI KJIacca K npHH3AJIe)KaT O,n;HOMY H TO-
MY)Ke tL1IeH)' KJIaCca L.
4. JII06bIM ,ZJ;ByM tL1IeHaM KJIacca L npHH3AJIe)KHT B TOq-
HOCTH O,ZJ;HH 06InHH JVIH HHX tL1IeH KJIacca K.
5. HH O,ZJ;HOMY qJIeH)' KJIacca L He npHH3AJIe)KHT 60JIee
qeM ,ZJ;Ba 3JIeMeHTa KJIacca K.
113 3Toro He60JIbmOrO nepeqHH nocryJIaTOB MbI MO)KeM,
nOJIb3YHCb 06bIqHbIMH npaBWIaMH JIOrHqeCKOrO BbIBo,n;a,
BbIBecTH HeCKOJIbKO TeopeM. HanpHMep, MO)KHO nOKa-
3aTb, qTO K cO,ZJ;ep)KHT B TOqHOCTH TpH tL1IeHa. Ho COB-
21
2. np06/\eMa HenpOTJ.1BOpe4J.1BOCTJ.1
MeCTHMa JIH ,n;aHHaH CHCTeMa nocryJIaToB? HeJIb3H JIH
H3 HHX nOJIyqHTb npoTHBOpeqHe? 9TOT Bonpoc pemaeT-
CH (oTpHQaTeJIbHo) C nOMOm;blO CJIe,n;yrom;eH MO,n;eJIH.
TIYCTb K eCTb KJIacc TOqeK, qJIeHaMH KOToporo HBJIHIOTCH
BepmHHbI HeKoToporo TpeyrOJIbHHKa, a L - KJIaCC OT-
pe3KoB npHMbIx, q;yeHaMH KOToporo HBJIHIOTCH CTOpOHbI
3Toro )Ke TpeyrOJIbHHKa. YCJIOBHMCH nOHHMaTb npe1VIo-
)KeHHe «q;yeH KJIacca K npHHa,zvre)KHT tt71eH)' KJIacca L»
KaK yrBep eHHe 0 TOM, qTO ,n;aHHaH TOqKa-BepmHHa
npHHa,zvre)KHT ,n;aHHOMY OTpe3KY-CTopOHe. TIpH TaKOM
u
nOHHMaHHH K bIH H3 nepeqHCJIeHHbIX nHTH nocryJIa-
TOB OKa3bIBaeTCH HCTHHHbIM yrBep eHHeM. HanpHMep,
u
nepBbIH nocryJIaT yrBep aeT Tor,n;a nonpocry, qTO JIIO-
6bIe ,n;Be TOqKH, HBJIHIO HeCH BepmHHaMH HeKOToporo
TpeyrOJIbHHKa, npHHa,zvre)KaT B TOqHOCTH O,n;HOM)' OTpe3-
KY, CJIY'Ka eMY CTOpOHOH 3Toro TpeyrOJIbHHKa. AHano-
rHqHbIM 06pa30M MbI y6e aeMCH B HCTHHHOCTH OCTanb-'
u u
HbIX nocryJIaTOB H B COBMeCTHMocm BceH ,n;aHHOH CHCTe-
MbI nocryJIaTOB B QeJIOM.
HenpOTHBOpeqHBOCTb reOMeTpHH PHMaHa TalOKe, OKa-
3bIBaeTCH, MO)KHO YCTaHOBHTb npH nOMOIIJ;H MO,n;eJIH, pe-
aJIH3YIOIIJ;eH ee nocryJIaTbI. MbI MO)KeM HHTepnpeTHpo-
BaTb (HCTOJIKOBaTb) CJIOBO «nJIOCKOCTb», <pHrypHpyro ee
B <POPMYJIHpOBKax pHMaHOBCKHX aKCHOM, KaK nOBepx-
HOCTb HeKOTopOH (eBKJIH,nOBOH!) C<pePbI, no,n; «TOqKOH»
nOHHMaTb TOqKY, JIe)Kam;yro Ha 3TOH c<pepHqeCKOH no-
BepXHOCTH, no,n; «npHMOH» - 1JYIY 60JIbmOrO Kpyra 3TOH
nOBepXHocm H T. n. Tor,n;a K bIH nocryJIaT pHMaHOB-
u u u
CKOH CHCTeMbI OKa3bIBaeTCH TeopeMOH eBKJIH,n;OBOH reo-
MeTpHH. CKIDKeM, pHMaHOB nocryJIaT napaJIJIeJIbHOCm
npH TaKOH HHTepnpeTaQHH rJIaCHT: «Qepe3 TOqKY, JIe)Ka-
22
2. np06AeMa HenpOTJIIBOpe4J11BOCTJII
Ha nOBepxHocTH c<I>epbI, HeJIb3H npoBecTH HH O,n;HOM
,n;yrH 60JIbmOrO Kpyra STOM c<I>epbI, KOTOpaH He nepece-
KaJIa 6bI npOH3BOJIbHOM ,n;aHHOM OKp)')KHOCTH 60JIbmOrO
Kpyra, BbI6paHHOM Ha STOM nOBepxHocTH».
Ho npHBe,n;eHHoe pacc eHHe He HBJIHeTCH HCqepnbIBa-
IOrnHM AOKa3aTeJIbCTBOM HenpOTHBOpeqHBOCTH reOMeT-
pHH PHMaHa: BeAb OHO cyrneCTBeHHO onHpaeTCH Ha AO-
nyrneHHe 0 HenpOTHBOpeqHBOCTH reOMeTpHH EBKJIHAa.
TaK qTO Tenepb HeH36e)KHO BCTaeT Bonpoc: a ,n;eMCTBH-
TeJIbHO JIH HenpOTHBOpeqHBa caMa reOMeTpHH EBKJIHAa?
no AaBHO YCTaHOBHBmeMCH TpClAHIJ:HH Ha TaKOM Bonpoc
OTBeqaJIH 06bIqHO B TOM A)'Xe, qTO «aKCHOMbI EBKJIH,n;a
HCTHHHbI, a CTaJIO 6bITb, H HenpOTHBOpeqHBbI». Ho Ta-
KOM OTBeT MbI )')Ke He MO)KeM 60JIee paCCMaTpHBaTb KaK
YAOBJIeTBOpHTeJIbHbIM eMbI erne BepHeMCH K STOM Te-
Me H pa3'bHCHHM nOAPo6Hee, B qeM HMeHHO 3aKJIlOqa-
eTCH ero HeYAOBJIeTBOpHTeJIbHOCTb). ,l(pyrOM OTBeT CO-
CTOHT B TOM, 1:JTO eBKJIHAOBCKHe aKCHOMbI COrJIacyroTCH
C <l>aKTHqeCKHMH - XOTH H orpaHHqeHHbIMH - ,n;aHHbIMH
Hamero onbITa H Ha6JIIOAeHHH, OTHOCHrnHMHCH K npo-
CTpaHCTBY, H qTO, rrpHHHMaH STH aKCHOMbI, MbI BnpaBe
o606 HTb, SKCTparrOJIHpOBaTb HamH 3HaHHH 0 HeKOTO-
pOM orpaHHqeHHOM 06JIaCTH. OAHaKO caMoe 60JIbmee,
Ha qTO MbI MO)KeM paCCqHTbIBaTb, HCXOAH H3 TaKHX «HH-
,nyKTHBHbIX» coo6pIDKeHHM, - TO, qTO aKCHOMbI «npaB-
AonOAo6HbI», HCTHHHbI «c 60JIbmOM BepOHTHOCTblO».
CJIeA)'lO HM BIDKHbIM mar B peIIieHHH 06c aeMOM 3AeCb
rrp06JIeMbI HerrpOTHBOpeqHBOCTH eBKJIH,n;OBOM reOMeT-
pHH npe,n;rrpHHHJI IHJIb6epT. OCHOBHaH HAeH ero MeTo,n;a
u u u
rro,n;CKa3aHa aHaJIHTHqeCKOH reOMeTpHeH, BOCXOAHrneH
erne K ,l(eKaprry. B npe,lVIO)KeHHOM IRnb6epToM «AeKap-
23
2. np06/\eMa HenpOTiltBOpe411tBOCTilt
u
TOBCKOH» HHTeprrpeTa HH eBKnH OBCKHX aKCHOM OHH
OqeBH HbIM o6pa30M CTaHOBHTCH HCTHHHbIMH anre6pa-
HqeCKHMH YTBep eHIDIMH. HarrpHMep, <pHrypHpyrom;ee
u
B aKCHOMax rrJIOCKOH reoMeTpHH CJIOBO «TOqKa» OJI)KHO
u
03HaqaTb Terrepb rrapy eHcTBHTenbHbIx qHCen, «rrpH-
MaH» - qHCJIOBOe cooTHomeHHe, BbIpIDKaeMoe ypaBHe-
u
HHeM rrepBoH CTerreHH C ByMH HeH3BecTHbIMH, «OKp)"K-
HOCTb» - qHCJIOBOe cooTHomeHHe, BbIpIDKaeMoe KB -
paTHbIM ypaBHeHHeM HeKoToporo crre HanbHoro BH a,
H T. . feOMeTpHqeCKOe rrpe,lVIO)KeHHe, rJIaCHm;ee, qTO Be
pa3JIHqHbIe TOqKH o H03HaqHbIM o6pa30M orrpeAeJIH-
lOT HeKOTOpyro rrpHMyro, rrepeXO HT Terrepb B HCTHHHoe
YTBep eHHe anre6pbI, COrJIaCHO KOTOPOMY Be pa3.7IHq-
u
HbIX rrapbI eHCTBHTenbHbIX qHCen O H03HaqHO orrpeAe-
u
JIHIOT HeKOTopoe JIHHeHHOe ypaBHeHHe; reOMeTpHqeCKaH
u
TeopeMa, COrJIaCHO KOTOpOH rrpHMaH H OKp)"KHOCTb rrepe-
ceKalOTCH He 60JIee qeM B ABYX TOqKax, rrepeXOAHT B an
re6paHqecKYlO TeopeMY 0 TOM, qTO CHCTeMa, COCTOH aH
u u
H3 JIHHeHHOro H KB paTl!oro ypaBHeHHH C AByMH He-
H3BeCTHbIMH, HMeeT caMoe 60JIbmee ABe rrapbI AeHCTBH-
TenbHbIX KopHeH, H T. A. Kopoqe rOBopH, HerrpoTHBope-
qHBOCTb eBKnH OBCKHX rrocryJIaTOB 060CHoBbIBaeTcH TeM
06CTOHTeJIbCTBOM, qTO OHH BbIrrOJIHHIOTCH Ha HeKOTopOH
anre6paHqeCKOH MO eJIH.
TaKoH MeTOA oKa3aTenbCTBa HerrpOTHBOpeqHBOCTH BeCbMa
IIJIo OTBOpeH H 3<p<peKTHBeH. Ho H rrpH 3TOM OCTalOTCH
BbICKa3aHHbIe BbIme B03pIDKeHIDI. B caMOM ene, Be b
H 3 ecb rrp06JIeMa, rrOCTaBJIeHHaH ,ltJIH OAHOH 06JIaCTH,
JIHmb rrepeBoAHTcH B APyryro 06JIaCTb. fwIb6epTOBCKoe
AOKa3aTenbcTBo HerrpoTHBOpeqHBOCTH ero CHCTeMbI reo-
MeTpHqeCKHX rrocryJIaTOB rroKa3bIBaeT, q'fO eCJIH «an-
re6pa» (TOqHee, apH<pMeTHKa AeHcTBHTenbHbIx qHcen)
24
2. np06AeMa HenpOTiltBOpe411tBOCTilt
HenpOTHBOpe"tlHBa, TO HenpOTHBOpe"tlHBa H 3Ta reOMeT-
pHH. JlCHO, "tITO AOKa3aTeJIbCTBO, c ecrBeHHo 3aBHCH-
m;ee OT npeAnOJIO>KeHHH 0 HenpOTHBOpe"tlHBOCTH HeKO-
TOpOH ApyrOH cHcreMbI, He HBJIHeTCH «a6COJIIOTHbIM» AO-
Ka3aTeJIbCTBOM HenpOTHBOpe"tlHBOCTH.
Bce nOnbITKH pemeHHH np06JIeMbI HenpOTHBOpe"tlHBOCTH
HaTaJIKHBaJIHCb Ha OAHO H TO >Ke 3aTpYAHeHHe: aKCHOMbI
u
HHTepnpeTHpOBaJIHCb C nOMOm;blO MOAeJIeH, COAep>Ka-
m;HX 6eCKOHe"tlHOe MHO>KeCTBO SJIeMeHTOB. BBHA)' 3Toro
HH OAIIY H3 TaKHX MOAeJIeH HeJIb3H 6bIJIO 0603peTb B KO-
He"tlHOe "tIHCJIO marOB, TaK "tITO HcrHHHOcrb aKCHOM Bce
em;e OCTaBaJIaCb nOA COMHeHHeM. HHA)'KTHBHoe pacc)')K-
AeHHe, 060CHOBbIBalOrn:ee HCTHHHOCTb eBKJIHAOBOH reo-
MeTpHH, HCnOJIb3yeT JIHmb KOHe"tlHOe "tIHCJIO Ha6JIIOAae-
MbIX <l>aKTOB, corJIacyrom;HXCH, nO-BHAHMOMY, C aKCHOMa-
MH. Ho 3aKJIIO"tIeHHe, no KOTOpOMY STa COrJIaCOBaHHOCTb
aKCHOM C Ha6JIIOAaeMbIMH <l>aKTaMH coxpaHHeT CBOIO CH-
JIY ,lVIH BceH 06JIaCTH H MO>KeT CJIy)KHTb onpaBAaHHeM
cHcreMbI aKCHOM B QeJIOM, caMO OCHOBaHO Ha 3KCTpano-
JIHQHH OT KOHe"tlHOrO K 6eCKOHe"tlHOMY.
KaKHM 06pa30M MO>KHO 6bIJIO 6bI 060CHOBaTb 3aKOH-
HOCTb CKa"tlKa "tIepe3 nponaCTb, OTAeJlHlOIIJ;yIO KOHe"tlHOe
OT 6eCKOHe"tlHOro? CJIe.lO'eT OTMeTHTb, "tITO ynOMHHYTaH
TpYAHOcrb YMeHbmaeTCH, - eCJIH H He COBceM YCTpa-
HHeTCH, - KorAa YAaeTCH nOCTpOHTb MOAeJIb, COCTOH-
IIJ;yIO JIHmb H3 KOHe"tlHOrO "tIHCJIa 3JIeMeHTOB. IlpHMepoM
u u
TaKOH KOHe"tlHOH MOAeJIH MO)KeT CJI)')KHTb onHCaHHaH
u
BbIme MOAeJIb-TpeyrOJIbHHK, nocpeACTBOM KOTOpOH MbI
YCTaHOBHJIH cOBMecrHMocrb nocrryJIaTOB, onHCbIBalOIIJ;HX
KJIaCCbI K H L. B TaKHX CJIy"tlaHX cpaBHHTeJIbHO JIerKO
<l>aKTH"tIeCKH npOBepHTb, AeHCTBHTeJIbHO JIH Bce 3JIeMeH-
25
2. np06/\eMa HenpOTiltBOpe411tBOCTilt
TbI MO,ZJ;eJIH Y,ZJ;oBJIeTBOpHIOT rrocTYJIaTaM, H TeM caMbIM
y6e,ZJ;HTbcH B HCTHHHOCTH (a 3HaqHT, H B COBMeCTHMO-
CTH) caMHX rrocTYJIaToB. CKIDKeM, HCTHHHOCTb rrepBoro
H3 yrrOMHHYTbIX TOJIbKO qTO rrocTYJIaTOB Y,ZJ;oCToBepHeTcH
TeM <paKToM, qTO qepe3 K bIe ,ZJ;Be BepmHHbI «MO,ZJ;eJIb-
u
Horo» TpeyrOJIbHHKa ,l];eHCTBHTeJIbHO rrpoxo,ZJ;HT B TOqHO-
CTH o,ZJ;Ha ero cTopoHa. TIOCKOJIbKY Bce 3JIeMeHTbI TaKoH
MO,ZJ;eJIH H HHTepecyroInHe Hac OTHOmeHHH Me)K,lJ;y HH-
MH ,ZJ;OC'fYIIHbI Herrocpe,ZJ;CTBeHHO H rrOJIHOM)' 0603peHHIO,
a orraCHOCTH ,ZJ;BYCMbICJIeHHOrO HCTOJIKOBaHHH pe3YJIbTa-
TOB TaKoro HCCJIe,ZJ;OBaHHH rrpaKTHqeCKH HeT, COBMeCTH-
MOCTb CHCTeMbI rrocryJIaTOB He MO)KeT 6bITb rrO,l];BeprHYTa
XOTb CKOJIbKO-HH6Y,ZJ;b 060CHOBaHHOM)' COMHeHHIO.
Ho, K CO)KaneHHIO, 60JIbmaH qaCTb CHCTeM rrOCTYJIaTOB,
HCrrOJIb3yeMbIx B KaqeCTBe OCHOBbI c eCTBeHHO BIDKHbIX
pa3,ZJ;eJIOB MaTeMaTHKH, He MO)KeT 6bITb HHTeprrpeTHpo-
BaHa C rrOMOInblO KOHeqHbIX MO,ZJ;eJIeH. TI03TOMY MbI HBHO
3axO,ZJ;HM B 'fYIIHK. KOHeqHbIe MO,ZJ;eJIH B rrpHHn:Hrre ,ZJ;O-
CTaTOqHbI JVIH YCTaHOBJIeHHH COBMeCTHMOCTH HeKOTopbIX
CHCTeM rrocryJIaTOB; HO 3TH CHCTeMbI HMelOT JVIH MaTeMa-
THKH BTopocrerreHHoe 3HaqeHHe. EeCKOHeqHbIe >Ke MO,ZJ;e-
JIH, Heo6xo,ZJ;HMbIe JVIH HHTeprrpeTan:HH 60JIbmeH qaCTH
Ba)KHbIX JVIH MaTeMaTHKH CHCTeM rrOCTYJIaTOB, MbI yMeeM
orrHCbIBaTb JIHmb B caMbIX 06 HX CJIOBax H He MO)KeM
u u
,l];aTb HHKaKOH TBep,ZJ;OH rapaHTHH, qTO TaKHe orrHcaHHH
caMH CBo6o,ZJ;HbI OT CKpbITbIX rrpOTHBOpeqHH.
KOHeqHO, XOTeJIOCb 6bI 6bITb YBepeHHbIMH B HerrpOTH-
BOpeqHBOCTH <POPMYJIHpOBOK, orrHCbIBalOInHX 6eCKOHeq-
HbIe MO,ZJ;eJIH, HO TaKHX, qTO Bce HCrrOJIb3yeMbIe HMH
OCHOBHbIe rrOHHTHH rrpe,ZJ;CTaBJIHIOTCH COBepmeHHO «HC-
HbIMH» H «OTtleTJIHBbIMH». Ho HCTOpHH HayKH He MO)KeT
26
2. np06AeMa HenpOTiltBOpe411tBOCTilt
nOXBaCTaTbCH TeM, qTO eM Be3JIO Ha AOKTpHHI,I, onepH-
pyrom:He HCKJIlOqHTeJII,HO HCHI,IMH H OTtleTJIHBI,IMH HAe-
u u
HMH H nOKOHm:HeCH Ha TBepAOH HHT)'HTHBHOH OCHOBe,
a HMeHHO Ha HHX H npHXOAHTCH AeJIaTI, BecI, paCqeT.
B HeKOTopI,IX 06JIaCTHX MaTeMaTHKH, ,lVIH KOTOpI,IX cym:e-
cTBeHHyIO pOJII, HrpalOT pa3JIHqHI,Ie Aonym:eHIDI 0 6ec-
KOHeqHI,IX COBOK)'IIH0 CTHX, 6I,IJIH 06Hap)')KeHI,I BeCI,Ma
cepI,e3HI,Ie npOTHBOpeqIDI, H 3TO HeCMOTpH Ha HHTYHTHB-
u
HyIO HCHOCTI, nOHHTHH, HCnOJII,3yeMI,Ix npH 3TOM, If Ka-
>Kym:yIOCH KOppeKTHOCTI, npHMeHHeMI,IX B AaHHI,IX Teo-
pIDIX YMCTBeHHI,IX KOHCTp)'KQHM. TaKHe npOTHBOpeqIDI
(HMeHYeMI,Ie 06I,IqHO «aHTHHOMIDIMH») 6I,IJIH o6Hapy-
)KeHI,I, B qaCTHOCTH, B nOCTpOeHHOM reoprOM KaHTopOM
B KOHQe XIX B. TeopHH 6eCKOHeqHI,IX MHO)KeCTB; npo-
THBOpeqIDI 3TH nOKa3aJIH, q'fO K aHCH HCHOCTI, Aa-
)Ke TaKoro 3JIeMeHTapHoro nOHHTIDI, KaK nOHHTHe MHO-
)KeCTBa (KJIaCca, COBOK)'IIHOCTH), He MO)KeT 06eCneqHTb
u u
HenpOTHBOpeqHBOCTH HH OAHOH KOHKpeTHOH CHCTeMI,I,
B KOTOpOM HCnOJII,3yeTCH TaKoe nOHHTHe. TIOCKOJII,KY )Ke
u
MaTeMaTHqeCKaH TeopIDI MHO)KeCTB, B KOTOpOH paCCMaT-
u u
pHBalOTCH CBOHCTBa COBOK)'IIHOCTeH 3JIeMeHTOB, qaCTO
u
npOB03rJIamaeTCH OCHOBOH ,lVIH OCTaJII,HI,IX pa3AeJIOB Ma-
TeMaTHKH (B qaCTHOCTH, 3JIeMeHTapHOM apH<pMeTHKH),
eCTeCTBeHHO cnpOCHTI" He npOHHKalOT JIH npOTHBope-
qIDI, nOAo6HI,Ie TeM, qTO 6I,IJIH 06Hap)')KeHI,I B <POPMY-
JIHpOBKe TeopHH 6eCKOHeqHI,IX MHO)KeCTB, H B APyrHe
MaTeMaTHqeCKHe AHCQHnJIHHI,I.
If B nOATBep eHHe TaKoro nOA03peHIDI EepTpaH Pac-
CeJI nOCTpOHJI npOTHBOpeqHe, OCTaBaHCI, HCKJIlOqHTeJII,-
u
HO B paMKax 3JIeMeHTapHOH JIOrHKH, - npOTHBOpeqHe,
B TOqHOCTH nOAo6Hoe TOMY, qTO 6I,IJIO 06Hap)')KeHO nep-
BOHaqaJII,HO B KaHTopOBCKOM TeopHH 6eCKOHeqHI,IX KJIac-
27
2. np06/\eMa HenpOTiltBOpe411tBOCTilt
COB (MHO:>KeCTB). AHTHHOMHIO PaCCeJIa MO)KHO OrrHCaTb
C'JIe IO HM 06pa30M. Ey,n;eM pa3JIHQaTb KJIaCCbI B 3aBH-
CHMOCTH OT TOrO, HBJIHIOTCH JIH OHH CBOHMH C06CTBeH-
HbIMH 3JIeMeHTaMH HJIH HeT. Ha30BeM KJIaCC «HOpMaJIb-
HbIM» B TOM H TOJIbKO B TOM C'JIyqae, KOr,n;a OH He co,n;ep-
:>KHT caMoro ce6H B KaqeCTBe 3JIeMeHTa; B rrpoTHBHOM )Ke
C'JIyqae 6y,n;eM Ha3bIBaTb KJIaCC «HeHOpMaJIbHbIM». TIpH-
MepoM HOpMaJIbHOrO KJIaCca MO)KeT C'JI)')KHTb KJIaCC Bcex
u
MaTeMaTHKOB - Be,n;b caM TaKOH KJIaCC He HBJIHeTCH,
OqeBH,n;HO, MaTeMaTHKOM H He HBJIHeTCH rrOTOMY CBOHM
co6CTBeHHbIM 3JIeMeHTOM. TIpHMepoM HeHOpMaJIbHOrO
KJIaCca HBJIHeTCH KJIaCC Bcex MbIC'JIHMbIX Be eH; caM 3TOT
u
KJIaCC HBJIHeTCH, OqeBH,n:HO, «MbIC'JIHMOH Be blO», a TeM
caMbIM - H CBOHM co6CTBeHHbIM 3JIeMeHTOM.
Orrpe,n;eJIHM Terrepb KJIaCC N - KJIacc Bcex HOpMaJIbHbIX
KJIaCCOB. JlBJIHeTCH JIH N HOpMaJIbHbIM KJIaCcoM? EC'JIH
N HOpMaJIeH, TO OH HBJIHeTCH CBOHM co6CTBeHHbIM 3JIe-
MeHTOM (Be,n;b, rro orrpe,n;eJIeHHIO, N co,n;ep)KHT Bee HOp-
MaJIbHbIe KJIaCCbI). Ho B TaKOM C'JIyqae N HeHOpMaJIeH,
TaK KaK B CRJIY ,n;aHHoro BbIllie orrpe,n;eJIeHIDI KJIaCC, co-
,n;ep:>Karn;HH caMoro ce6H B KaqeCTBe 3JIeMeHTa, HBJIHeTCH
HeHOpMaJIbHbIM. C ,n;pyrOH CTOpOHbI, eC'JIH N - HeHop-
MaJIbHbIH KJIaCC, TO OH (B CHJIY orrpe,n;e.rreHIDI rrOHHTIDI
HeHOpMaJIbHOCTH) HBJIHeTCH CBOHM co6CTBeHIlbIM 3JIe-
MeHTOM; HO B TaKOM C'JIyqae N HOpMaJIeH, TaK KaK BbIllie
orrpe,n;eJIeHO, qTO 3JIeMeHTaMH N HBJIHIOTCH JIHllib HOp-
MaJIbHbIe KJIaCCbI. Kopoqe rOBopH, N HOpMaJIeH Tor,n;a
H TOJIbKO Tor,n;a, Kor,n;a N HeHOpMaJIeH. OTclO,n;a C'JIe eT,
qTO yrBep)K,lJ;eHHe «N - HOpMaJIbHbIH KJIacc» HBJIHeTCH
B O,n;HO H TO :>Ke BpeMH HCTHHHbIM H JIO)KHbIM. 9TO rrpoTH-
BOpeQHe HeMHH)'eMo C'JIe eT H3 HeKpHTHqeCKOrO, 6e30-
rOBOpOqHOrO yrrOTpe6JIeHIDI rrpe,ZJ;CTaBJIHlOrn;erOCH CTOJIb
28
2. np06AeMa HenpOTiltBOpe411tBOCTilt
HCHbIM rrOHHTHH KJIaCca (MHO)KeCTBa). BrrOC'JIe,n;cTBHH 6bI-
JIH 06Hap)')KeHbI H ,n;pyrHe rrapa,n;oKcbI, rrpHqeM Ka bIH
H3 HHX CTpORJICH C rroMOm;blO xopomo H3BeCTHbIX H BpO-
,n;e 6bI 6eccrropHbIx rrpHeMoB pacc)')K,ZJ;eHHH. MaTeMaTH-
KaM rrpHIlL7IOCb rrpHHTH K BbIBO,n;y, qTO rrpH rrocTpoeHHH
rrpeTeH,n;yrom;HX Ha HerrpoTHBOpeqHBOCTb CHCTeM o6m;e-
u
H3BeCTHOCTb H HHTYHTHBHaH HCHOCTb H,n;eH HBJIHIOTCH ,n;a-
u u
JIeKO He Ha,n;e)KHOH OCHOBOH.
MbI y6e,n;RJIHCb B Ba)KHOCTH rrp06JIeMbI HerrpOTHBOpeqH-
BOCTH (COBMeCTHMOCTH) H 03HaKOMRJIHCb C KJIaCCHqe-
CKHM, «CTaH,n;apTHbIM», MeTo,n;OM ee pemeHHH C rrOMO-
m;blO MOAeJIeH. MbI BHAeJIH, qTO rrp06JIeMa 3Ta 06bIqHO
Tpe6yeT HCrrOJIb30BaHHH 6eCKOHeqHbIX MOAeJIeH, onHca-
HHe KOTOpbIX, o,n;HaKO, caMO qpeBaTO BH)'TpeHHHMH npo-
THBOpeqHHMH. HaM rrpHAeTcH COrJIaCHTbCH rro3ToMY, TO
MeToA MOAeJIeH HMeeT orpaHHqeHH)'IO n;eHHocTb B Kaqe-
CTBe OPYAHH pemeHHH np06JIeMbI H HeAOCTaTOqeH ,lVIH
rrOJIyqeHHH OKOHqaTeJIbHOrO OTBeTa Ha Hee.
3
A6cOAIOTHbie AOKa3aTeAbCTBa
HenpOTHBOpe HBOCTH
ITpHHU;HnHaJIbHbIe OrpaHHqeHIDI, npen.HTCTByroIIJ;He HC-
v
nOJIb30BaHHIO MO eJIeH AJI.H YCTaHOBJIeHIDI HenpOTMBO-
peqHBOCTH H nepepaCTalOIIJ;He B )'BepeHHOCTb nO 03pe-
HIDI, q'f0 MHOrne MaTeMaTHqeCKHe CHCTeMbI qpeBaTbI
BHYTPeHHHMH npOTHBOpeqIDIMH, npHBeJIH K TOMY, qTO
6bIJIH npe npHIDITbI COBepmeHHO HOBbIe nonbITKH pe-
meHIDI np06JIeMbI HenpOTHBOpeqHBOCTH. MbTepHaTHB-
v
HbIH - no OTHomeHHIO K ynoMHHaBmHMC.H O CHX nop
v
OKa3aTeJIbCTBaM OTHOCHTeJIbHOH HenpOTHBOpeqHBOCTH -
no xo 6bIJI YKa3aH fwIb6epTOM. Ero U;eJIblO 6bIJIO no-
CTpoeHHe «a6COJIIOTHbIX» OKa3aTeJIbCTB HenpOTHBope-
qHBOCTM pa3JIHqHbIX CHCTeM - OKa3aTeJIbCTB, He HC-
v
XOM HX H3 npe nOJIO)KeHHH 0 HenpOTHBOpeqHBOCTH
KaKOH-JIH60 pyrOH CHCTeMbI. QT06bI nOIDITb CYIIJ;HOCTb
OTKpbITIDI re eJI.H, HaM nOHM06HTC.H pa306paTbC.H B 06-
HX qepTax B rHJIb6epToBcKOM' no xo e K np06JIeMe.
ITepBbIM maroM nOCTpoeHIDI a6COJIIOTHOrO OKa3aTeJIb-
CTBa HenpOTHBOpeqHBOCTH, COrJIaCHO TaKOMY noAXO
OJI)KHa .HBHTbC.H nOIlHaR ifjoPMWlU3a14Uf1. HCCJIe eMOH
31
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBope4HBocTH
e THBHOH cHcTeMbI, COCTO.Hm;eH, rpy60 rOBOp.H, B TOM,
qTO BCe BXOMm;He B aHH)'IO, CHCTeM)' BbIpIDKeHIDI pac-
CMaTpHBaIOTC.H KaK JIHmeHHbIe KaKoro 6bI TO HH 6bIJIO
3HaqeHIDI - npoCTO KaK HeKOTopbIe COqeTaHIDI CHM-
BOJIOB. Cnoc06bI coe HHeHIDI CHMBOJIOB H 06pam;eHIDI
C COCTaBJIeHHbIMH H3 HHX BbIpIDKeHIDIMH qeTKO npe-
CMOTpeHbI Cne HaJIbHbIMH npaBHJIaMH. B pe3YJIbTa-
Te MbI nOJIyqaeM CHCTeMY CHMBOJIOB (Ha3bIBaeMyro «HC-
qHCJIeHHeM»), Co ep>Ka Bce Te H TOJIbKO Te CHMBO-
JIbI, Ha KOTopbIe MbI .HBHbIM H He BYCMbICJIeHHbIM 06pa-
30M YKa3aJIH. IToCT)'JIaTbI H TeopeMbI nOJIHOCTblO <pop-
MaJIH30BaHHOH CHCTeMbI - npoCTO «CTpOqKH» (T. e. KO-
HeqHbIe nOCJIe OBaTeJIbHOCTH) HHqerO He 03HaqalOm;HX
3HaqKOB, OCTpoeHHbIe H3 3JIeMeHTapHbIX CHMBOJIOB CO-
rJIaCHO npaBHJIaM aHHOH CHCTeMbI. B TaKOH nOJIHOCTbIO
<pOpMaJIH30BaHHOH CHCTeMe BbIBO TeopeM H3 nocryJIa-
TOB - He qTO HHoe, KaK npe06pa30BaHHe (COrJIaCHO npa-
BHJlaM CHCTeMbI) O HOH COBOK)'IIHOCTH «CTpOqeK» B PY-
ryro. ITOCT}'IIaH TaKHM 06pa30M, MbI H36eraeM onaCHO-
u
. CTH, CB.H3aHHOH C He.HBHbIM HCnOJIb30BaHHeM KaKHX-JIH-
60 COMHHTeJIbHbIX MeTO OB pacc eHIDI.
<l>opMaJIH3a IDI - eJIO OBOJIbHO-TaKH TpY Hoe H Tpe-
6yrom;ee HeMaJIOH H306peTaTeJIbHOCTH; HO OHa xopomo
CJI}')KHT HaMeqeHHOH 3Maqe. <l>opMaJIH3a IDI n03BOJI.H-
eT .HCHO BH eTb CTpYKTYPY CHCTeMbI H Ha3HaqeHHe OT-
eJIbHbIX ee 3JIeMeHTOB aHaJIOrHqHO TOM)', KaK CTpyKTy-
pa H pa60Ta OT eJIbHbIX Y3JIOB KaKOH-HH6Y b MamHHbI
u
JIerqe Y.HCHHIOTC.H Ha MO eJIH TaKOH MamHHbI, qeM npH
paccMoTpeHHH caMoH MamHHbI. JIorINeCKHe COOTHome-
HIDI Me OT eJIbHbIMH npeAJIO>KeHIDIMH CTaHOB.HTC.H
nOCJIe <popMaJIH3a HH xopomo 0603pHMbIMH; MbI B HM
u
B HeH CTpYKTYPHbIe COOTHomeHIDI Me pa3JIINHbIMH
32
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBope4HBocTH
«CTpOqKaMH» H «6ecCMbICJIeHHbIMH» CHMBOJIaMH, YHC-
HHeM, KaKHM 06pa30M OHH CBH3aHbI ,npyr C ,npyrOM, IIpa-
BHJIa HX KOM6HHaIJ;HH H B3aHMHoro CJIe,nOBaHHH H T. II.
O CHX IIOp MbI roBOpHJIH, qTO «6ecCMbICJIeHHbIe» 3HaqKH
TaKoH <pOpMaJIH30BaHHoH MaTeMaTHKH HHqerO He ymeep-
:JICaalOm - 3TO IIOKa IIpOCTO HeKaH a6CTpaKTHaH Kap-
u
THHKa, HJIJIIOCTpHpyro aH CTpoeHHe HHTepecyro eH Hac
CHCTeMbI. Ho, KOHeqHO, CTpoeHHe TaKOH KapTHHKH -
u
a TeM caMbIM H HJIJIIOCTpHpyeMoH elO CHCTeMbI - MbI
MO)KeM OIIHCbIBaTb Ha 06bIqHOM qe.nOBeqeCKOM H3bIKe,
,neJIaH OIIpe,ne.rreHHbIe BbICKa3bIBaHHH, OTHOCH HeCH K ee
06 eH KOH<pHrypaIJ;HH H COOTHomeHHHM OT,ne.nbHbIX ee
3JIeMeHTOB.
MbI MO)KeM, HaIIpHMep, OTMeTHTb IIpOCTOry HJIH CHMMeT-
PHqHOCTb KaKoH-HH6y,nb «CTpOqKH», CXO,nCTBO ee C HeKO-
u u u
TOpOH ,npyrOH «CTpOqKOH» MO)KeM 3aMeTHTb, qTO o,nHa
«CTpOqKa» MO)KeT 6bITb IIpe,nCTaBJIeHa B BH,ne COqJIeHe-
HHH Tpex ,npyrHX «CTpOqeK» H T. II. TaKHe BbICKa3bIBaHHH,
6e3YCJIOBHO, OCMbICJIeHbI H, 60JIee Toro, MO:ryr BbIpIDKaTb
BeCbMa c eCTBeHH)'IO HH<popMaIJ;HIO 0 HameH <pOpMaJIb-
HOH CHCTeMe. CJIe.nyeT, o,nHaKO, cpa3Y )Ke OTMeTHTb, qTO
Bce 3TH OCMbICJIeHHbIe BbICKa3bIBaHIDI 0 6eCCMbICJIeHHOH
(HJIH, qTO TO )Ke caMoe, - <pOpMaJIH30BaHHOH) MaTeMa-
THKe HHKOHM 06pa30M He IIpHHa.rvIe)KaT caMH IIO ce6e
3TOR MaTeMaTHKe. OHH OTHOCHTCH K 06JIaCTH, KOTOpyro
fwIb6epT Ha3BaJI «MeTaMaTeMaTHKoH», K H3bIKY, Ha KO-
mopOM rOBopHT 0 MaTeMaTHKe. MeTaMaTeMaTHqeCKHe
BbICKa3bIBaHIDI - 3TO BbICKa3bIBaHIDI 0 CHMBOJIaX, BXO-
M HX B <pOpMaJIH30BaHH)'IO MaTeMaTHqecK)'IO CHCTeMY
(T. e. B HCqHCJIeHHe), 0 BH,nax CHMBOJIOB, 06 HX yno-
p OqeHHH BHyrpH <pOpMaJIbHOR CHCTeMbI, 0 cIIoc06ax
33
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBope4HBoCTH
COCTaBJIeHIDI H3 3THX CHMBOJIOB 60JIee ,lVIHHHbIX 3Ha-
KOCOqeTaHHH (<<CTpOqeK»), KOTopbIe eCTeCTBeHHO Ha3bI-
BaTb «<popMYJIaMH» cHCTeMbI, HaKoHen;, 0 COOTHomeHIDIX
Me <popMYJIaMH, B qaCTHOCTH 0 TOM, KaKHe <POPMY-
JIbI MOIyr 6bITb nOJIyqeHbI (no <pHKCHpOBaHHbIM HaMH
npaBHJIaM 06pam;eHHH C HHMH) B KaqeCTBe «CJIe,nCTBHH»
,npyrHX <POPMYJI.
ITpHBe,neM HeCKOJIbKO npHMepOB, HJIJIIOCTpHpyrom;HX pa3-
JIHqHe Me MaTeMaTHKOH H MeTaMaTeMaTHKOH. CKa-
)ReM, BbIpa)ReHHe «2 + 3 =.5» npHHa,nJIe)RHT MaTeMa-
THKe (apH<pMeTHKe) H CTpOHTCH HCXO,lUl JIHmb H3 3JIe-
MeHTapHbIx apH<pMeTHqeCKHX CHMBOJIOB, B TO BpeMH KaK
BbICKa3bIBaHHe «" 2 + 3 = 5" eCTb apH<pMeTHqeCKaH <pop-
MYJIa» yrBep aeT HeqTO 06 3TOM BbIpIDKeHHH. OHO ca-
MO no ce6e He BbIpIDKaer HHKaKoro apH<pMeTHqeCKOrO
<paKTa H He npHHa,nJIe)RHT <pOpMaJIbHOMY H3bIK)' apH<p-
MeTHKH, a OTHOCHTCH K MeTaMaTeMaTHKe, xapaKTepH3YH
HeKOTOpyro CTpOqK)', COCTaBJIeHH)'IO H3 apH<pMemqeCKHX
CHMBOJIOB, KaK <POPMY
<l>OPMYJIbI
x = x,
0=0,
0#0
npHHa,nJIe)RaT MaTeMaTHKe (apH<pMeTHKe); K aH H3 HHX
COCTaBJIeHa H3 O,nHHX TOJIbKO apH<pMeTHqeCKHX 3HaKOB.
BbICKa3bIBaHHe )ICe «"x" eCTb nepeMeHHaH» OTHOCHTCH
}')Ke K MeTaMaTeMaTHKe, nOCKOJIbK)' OHO xapaKTepH3Y-
eT HeKOTopbIH apH<pMeTHqeCKHH CHMBOJI, yrBep aH,
qTQt OH npHHa,nJIe)RHT HeKOTopOMY Cnen;HaJIbHOMY KJIac-
cy CHMBOJIOB (a HMeHHO, KJIaccy nepeMeHHbIX). ITpH-
34
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpOTIt1BOpe41t1BOCTIt1
Ha.zt1Ie>KHT MeTaMaTeMaTHKe H BblCKa3bIBaHHe «<poPMY-
JIa ,,0 = 0" BbIBO HMa H3 <POPMYJIbI "x = x" nocpe -
CTBOM nO CTaHOBKH ,,0" BMeCTO nepeMeHHoH "x"», onH-
CbIBalOIIJ;ee onpe eJIeHHOe OTHomeHHe Me HeKoTopbI-
MH BYMH <popMYJIaMH. OTHOCHTCH K MeTaMaTeMaTHKe
H yrBep>KAeHHe «" 0 i= 0" He eCTb TeopeMa», rJIaCHIIJ;ee,
qTO HeKOTOpaH apH<pMeTlNeCKaH cpopMYJIa He MO>KeT
6bITb BbIBe eHa H3 aKCHOM apH<pMeTHKH. MeTaMaTeMa-
THKe, KOHeqHO, npHHa.zt1Ie>KHT H BbICKa3bIBaHHe «apH<p-
MeTHKa HenpOTHBOpeqHBa» (HHbIMH CJIOBaMH, H3 aKCH-
OM apH<pMeTHKH HeJIb3H BbIBeCTH BYX B3aHMHO npo-
THBOpeqHBbIX <POPMYJI, HanpHMep <POPMYJI « 0 = 0 »
H «0 i= 0»). JlCHO, qTO 3TO BbICKa3bIBaHHe rJIaCHT HeqTO
06 apH<pMeTHKe, a HMeHHO, OHO yrBep>KAaeT, qTO napbI
apH<pMeTHqeCKHX <poPMYJI onpe eJIeHHOrO BH a He Ha-
XOMTCH B onpe eJIeHHOM OTHomeHHH K <pOPMYJIaM, CO-
CTaBJIHIOIIJ;HM CHCTeMY aKCHOM apH<pMeTHKH.
C/\eAYeT OTMeTIt1Tb, 4TO Bce 3TIt1 MeTaMaTeMaTIt14eCKIt1e BbICKa3bIBaHIt151 He CO- +
Aep)l(aT HIt1KaKIt1X MaTeMaTI1L1eCKI1X 3HaKOB 11 CPOPMY/\, a COAep)l(aT /\It1Wb
It1X I1MeHa. Pa3/\It141t1e Me)I(AY Bblpa)l(eHIt151MIt1 It1 It1MeHaMIt1 Bblpa)l(eHIt1t:1 04eHb
BCDKHO. KcraTIt1, It1 B 06bl4HOM pa3rOBopHOM 5I3blKe HIt1KaKOe npeA/\O)l(eHlt1e
He COAep)l(lt1T 06beKTOB, 0 KOTOpblX B HeM rOBOpIt1TC5I, - OHO COAep)l(lt1T
/\It1Wb It1X It1MeHa. CKa)l(eM, KorAa Mbl rOBOplt1M 0 KaKOM-HIt16YAb ropOAe,
TO Mbl BcraB/\5IeM B npeA/\O)l(eHlt1e He caM ropOA, a ero It1M5I (Ha3BaHlt1e).
T04HO TaK )l(e, ec/\lt1 Mbl XOTIt1M CKa3aTb 4TO-HIt16YAb 0 KaKOM-/\1t160 C/\OBe
(1t1/\1t1 Boo6we /\I06oM 5I3bIKOBOM BblpCDKeHIt1It1), TO Mbl AO/\)I(Hbl It1CnO/\b30-
BaTb B Ka4ecrBe 4/\eHa npeA/\O)l(eHIt151 He caMO C/\OBO/Bblpa)l(eHlt1e, a ero
It1M5I. 06bl4HO 3TO Ae/\aeTC nplt1 nOMOWIt1 KaB 14eK. Hawe 1t13/\O)l(eHIt1e KaK
pa3 It1 c/\eAyeT 3TOMY 06b14alO. Mbl MO)l(eM CKa3aTb, Hanplt1Mep, «4It1Ka-
ro - 60/\bW0t:1 ropOA». Ho cppa3a «41t1KarO COCTOIt1T 1t13 TpeK c/\orOB»
6eCCMbic/\eHHa (6e3rpaMOTHa). 4T06bl Bblpa31t1Tb noc/\eAHee yrBep)I(AeHlt1e
npaBIt1/\bHO, Mbl AO/\)I(Hbl Hanlt1CaTb: «" 41t1KarO" COCTOIt1T 1t13 Tpex c/\orOB».
35
3. A6coJ\IOTHble AOKa3aTeJ\bCTBa HenpoTHBope4HBocTH
T04HO TaK )f(e HeBepHo 6blJ\O 6bl HanHcaTb:
«x = 5 eCTb ypaBHeHHe».
npaBHJ\bHa51 3anHCb TaKOBa:
«" X = 5" eCTb ypaBHeHHe».
KOHeqHO, pa3JIHqHe Me TeopHeH H MeTaTeopHeH MO-
>KeT OTHOCHTbC.H He TOJIbKO K MaTeMaTHKe - Be,nb 3TO
npOCTO xopomo H3BeCTHoe BceM HaM pa3JIHqHe Me>K-
l1Y KaKHM-JIH60 H3yqaeMbIM HaMH npe,nMeTOM H pa3ro-
BopaMH 06 3TOM npe,nMeTe. HanpHMep, BbICKa3bIBaHHe
u
«y nTHQ H3 po,na nJIaB}'HqHKOB .HHQa BbICH>KHBaIOT caM-
QbI» OTHOCHTCH K npe,nMery, H3yqaeMOMY 300JIOraMH,
H npHHa,zvIe>KHT 300JIOrHH; HO eCJIH MbI CKIDKeM, qTO
yrBep eHHe OTHOCHTeJIbHO n.naB}'HqHKOB nOKa3bIBaeT,
qTO B 300JIOrHH eCTb MHoro 3arMOqHOrO, TO 3TO )'>Ke 6y-
,neT yrBep eHHe He 0 nJIaB}'HqHKax, a 0 npe,nb M
BbICKa3bIBaHHH, H ,nHCQHnJIHHY, B KOTOpYID BXO,nHT TaKoe
c eHHe, CJIe,nOBaJIO 6bI Ha3BaTb MeTa300JIOrHeH.
TOqHO TaKOBO >Ke COOTHomeHHe Me MaTeMaTHKOH
u u
H MeTaMaTeMaTHKOH: npe,nMeT nepBoH COCTaB.1.UIlOT caMH
cpOpMaJIbHbIe cHcTeMbI 1 KOTopbIe npH1JYMbIBalOT MaTeMa-
THKH, npe,nMeT BTOpOH - onHcaHHe TaKHX cpOpMaJIbHbIX
cHcTeM, BbI.HCHeHHe H 06c eHHe HX CBOHCTB.
Ba>KHOCTb CTOJIb HaCTOHTeJIbHO no,nqepKHBaeMoro HaMH
pa3JIHqeHIDI MaTeMaTHKH H MeTaMaTeMaTHKH Tpy,nHO ne-
peOQeHHTb. MmopHpOBaHHe HJIH He,nooQeHKa 3Toro pa3-
JIHqeHIDI npHBOMT K He,nopa3)'MeHIDIM, a TO H K np.HMbIM
npOTHBOpeqIDIM. OC03HaHHe ero BIDKHOCTH n03BOJIHJIO
rJIy6>Ke Y.HCHHTb JIOrHqecKYIO CTpYKTYPY MaTeMaTHqe-
CKHX MeTO,nOB pacc eHIDI H qeTKO perJIaMeHTHpOBaTb
36
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBope4HBocTH
yrrOTpe6JIeHHe pa3JIHqHbIX <pOpMaJIbHbIX CHMBOJIOB, npe-
BpaIIJ;aH MaTeMaTHKY B qHCTO <pOpMaJIbHOe HCqHCJIeHHe,
cB060 Hoe OT BCHqeCKHX HeHBHO no pa3YMeBaeMbIx o-
nym;eHHM H n060qHbIX CMbICJIOBbIX aCCOIJ;HaIJ;HM. TOJIbKO
Ha 6a3e TaKHX HOBbIX npe CTaBJIeHHM CTaJIO B03MO)KHbIM
u
aTb TOqHbIe onpe eJIeHHH MaTeMaTHqeCKHX OnepaIJ;HH
H JIOrHqeCKHX npaBHJI, KOTOpbIMH MaTeMaTHKH nOJIb30-
BaJIHCb o Tex nop 6e3 HCHoro nOHHMaHIDI Toro, qTO )Ke,
c06cTBeHHo, OHH eJIaIOT.
fwIb6epT YJIOBHJI caMyro Cyrb np06JIeMbI, nOJIO)KHB B
OCHOBY CBOHX nonbITOK nocTpoeHHH «a6COJIIOTHbIX» o-
Ka3aTeJIbCTB HenpoTHBOpeqHBOCTH pa3JIHqHe Me <pop-
MaJIbHbIM HCqHCJIeHHeM Hero onHcaHHeM. OH nOCTaBHJI
3Maqy pa3BHTIDI CneIJ;HaJIbHOrO MeTo a, C nOMOIIJ;blO
KOToporo MO)KHO 6bIJIO 6bI npOBO HTb OKa3aTeJIbCTBa
HenpoTHBOpeqHBOCTH TOM )Ke CTeneHH y6e HTeJIbHOCTH,
qTO H OKa3aTeJIbCTBa, HCnOJIb3YIOIIJ;He KOHeqHbIe Mo e-
JIH, Ha KOTOpbIX peaJIH3yroTcH onpe eJIeHHbIe CHCTeMbI
nocTYJIaToB. MCKOMbIM MeTo OJI)KeH 6bIJI 6bI COCTOHTb
B HCqepnbIBalOIIJ;eM aHaJIH3e KOHeqHOrO qHCJIa cTpYKTYP-
HbIX CBOMCTB BbIpa)KeHHM B nOJIHOCTblO <pOpMaJIH30BaH-
HbIX HCqHCJIeHIDIX. AHaJIH3 OJI)KeH HCXO HTb H3 TOqHOH
<pHKCaIJ;HH pa3JIHqHbIX BH OB, BXOMIIJ;HX B paccMaTpH-
BaeMoe HCqHCJIeHHe CHMBOJIOB, YKa3aHIDI Ha cnoc06bI
coe HHeHIDI 3TH X CHMBOJIOB B <POPMYJIbI, onHcaHIDI cno-
c06a BbIBo a O HHX <POPMYJI H3 pyrHX H aBaTb cnoc06
pemeHIDI Bonpoca, BbIBO HMbI JIH <POPMYJIbI KaKOrO-JIH-
60 onpe eJIeHHOrO BH a H3 HeKOTopbIX onpe eJIeHHbIX
<POPMYJI nocpe CTBOM HBHO C<POPMYJIHpOBaHHbIx npaBHJI
onepHpOBaHIDI C <popMYJIaMH. fwIb6epT 6bIJI y6e eH
B TOM, qTO K oe MaTeMaTHqeCKOe HCqHCJIeHHe MO)KHO
u
npe CTaBHTb «Ha reOMeTpHqeCKHH MaHep», T. e. B BH e
37
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBopeLfHBoCTH
TaKOH COBOK)'IIHOCTH <POPMYJI, Ka aH H3 KOTOpbIX CBH-
3aHa C JII060H pyrOH <pOpM)'JIOH Toro >Ke HCqHCJIeHIDI
JIHIlIb CTp}'KTYPHbIMH COOTHOllieHIDIMH H3 HeKoToporo
u
KOHeqHOrO nepeqH.H COOTHOIlIeHHH.
Ha 3TOM y6e eHHH H OCHOBbIBaJICH ero paCqeT, qTO OH
cyMeeT nocpe CTBoM CHCTeMarnqeCKOrO H HCqepnbIBalO-
ero 0603peHIDI 3THX CTpYKTYPHbIX CBOHCTB BbIpIDKeHHH
u
aHHoH CHCTeMbI nOKa3aTb, qrrO H3 aKCHOM ,naHHoro HC-
qHCJIeHIDI HeJIb3H nOJIyqHTb <pOpMaJIbHO npOrnBOpeqa-
He pyr pyry <POPMYJIbI. eCTBeHHeHillHM YCJIOBH-
eM rHJIb6epTOBCKOH nporpaMMbI B nepBOHaqaJIbHOH ee
<POPMYJIHpOBKe 6bIJIO pa3peIlIeHHe ynOTpe6J1.HTb B OKa-
3aTeJIbCTBax HenpOTHBOpeqHBOCTH JIHillb TaKHe npHeMbI
pacc eHHH, KOTopbIe HH B KaKOH <popMe He HCnOJIb3Y-
lOT HH 6eCKOHeqHOrO MHO>KeCTBa CTp}'KTYPHbIX CBOHCTB
<POPMYJI, HH 6eCKOHeqHOrO MHO>KeCTBa onepa HH HM
<popMYJIaMH.
TaKHe MeTO bI pacc eHHH OH Ha3BaJI «<pHHHTHbIMH»,
a OKa3aTeJIbCTBa HenpOTHBOpeqHBOCTH, npOBe eHHbIe
<pHHHTHbIMH cpe CTBaMH, - «a6COJIIOTHbIMH». «A6co-
u
JIIOTHoe» OKa3aTeJIbCTBO OCTHraeT CBoeH eJIH C nOMO-
blO HeKOToporo MHHHMaJIbHOrO apCeHaJIa npHH HnOB
BWBo a H He HCXO HT H3 HenpOTHBOpeqHBOCTH HHKa-
KOH PyroH CHCTeMbI aKCHOM. TaKHM 06pa30M, eCJIH 6bI,
HanpHMep, Y aJIOCb nOJIyqHTb a6COJIIOTHOe OKa3aTeJIb-
CTBO HenpOTHBOpeqHBOCTH apH<pMeTHKH, OHO OJI>KHO
6bIJIO 6bI nocpe CTBOM HeKOTopOH <pHHHTHOH MeTaMa-
u
TeMaTHqeCKOH npo eAYPbI YCTaHOBHTb HeB03MO>KHOCTb
O HOBpeMeHHoro BbIBo a H3 aKCHOM (T. e. H3 HCXO -
HbIX <POPMYJI) apH<pMeTHKH C nOMO blO <pHKCHpOBaH-
u
HbIX npaBHJI BbIBo a HHKaKOH napbI B3aHMHO npOTHBO-
38
3. A6co/\IOTHbie AOKa3aTe/\bCTBa HenpoTHBope4HBocTH
peqHBbIX <POPMYJI, CKIDKeM «0 = 0» H ee oTpHIJ;aHIDI
«f'J (0 = 0)>> (3 ecb 3HaK «f'J» 03HaqaeT «He»).
BnO/\He T04HbiX YKa3aHHt:1 Ha TO, KaKHe HMeHHO MaTeMaTH4eCKHe MeTO- +
Abl c/\eAYeT C4HTaTb «cpHHHTHbIMH», rH/\b6epT He Aa/\. B nepBOHa4a/\b-
HOt:1 CPOPMY/\HpOBKe ero nporpaMMbl Tpe6oBaHHS1, KOTOpblM AO/\)f(Hbl 6bl/\H
YAOB/\eTBopS1Tb a6CO/\IOTHbie AOKa3aTe/\bCTBa HenpOTHBope4HBOCTH, 6bl/\H
3Ha4HTe/\bHO 6o/\ee CH/\bHbIMH, 4eM B noc/\eAYIOWHX pa3bS1CHeHHS1X rH/\b-
6epToBcKot:1 nporpaMMbl, AaHHblX npeACTaBHTe/\S1M WKO/\bl rH/\b6epTa.
Ey eT, nO)KanyH, He6eCIIOJIe3HO cpaBHHTb MeTaMaTeMa-
THKY, IIOHHMaeMyro KaK TeopHIO OKa3aTe.nbCTBa, C Teo-
pHeH maxMaTHoH HrpbI. B maxMaTbI HrpalOT C IIOMO-
IIJ;blO 32 <pHryp oIIpe e.neHHoro BHAa, IIepe BHralOIIJ;HXCH
no KBMpaTHoH OCKe, pa3 e.neHHoii Ha 64 KJIeTKH, IIPH-
qeM nepe BH)KeHIDI 3TH (<<XO bI») COBepmalOTCH IIO He-
KOTOpbIM CTporo OIIpe e.neHHbIM npaBHJIaM. Pa3YMeeT-
CH, ,nIDI HrpbI He Tpe6yeTcH HHKaKOH «HHTepIIpeTaIJ;HM»
<pHryp H HX pa3JIHqHbIX nOJIO)KeHHH Ha OCKe, XOTH Ta-
K)'IO HHTepIIpeTaIJ;HIO npH )Ke.naHHH MO)KHO 6bIJIO 6bI
H IIp aTb. HaIIpHMep, MO)KHO 6bIJIO 6bI CqHTaTb,
u
qTO nemKH - 3TO apMeHCKHe nOJIKH, a KJIeTKH OC-
KH - onpe e.neHHbIe reorpa<pHqeCKHe paHOHbI H T. II.
Ho TaKoro po a COrJIameHIDI (HHTepIIpeTaIJ;HH) He yno-
Tpe6HTe.nbHbI - Ha caMOM e.ne HH <pHrypbI, HH KJIeT-
KH OCKH, HH nOJIO)KeHIDI <pHryp He 03HaqalOT pOBHO
HHqerO BHe HrpbI KaK TaKOBOH. MHaqe rOBopH, MO)KHO
6bIJIO 6bI CKa3aTb, qTO <pHrypbI H HX IIOJIO)KeHIDI Ha OC-
Ke «6eCCMbICJIeHHbI». TaKHM 06pa30M, Hrpa B maxMa-
TbI HBJIHeTCH aJIeKO HM aHaJIOrOM <pOpMaJIH30BaH-
Horo MaTeMaTHqeCKOrO HCqHCJIeHIDI. <PHrypbI H KJIeTKH
OCKH COOTBeTCTByroT 3JIeMeHTapHbIM CHMBOJIaM HCqHC-
JIeHIDI; onYCTHMbIe IIpaBR1IaMH HrpbI II03HIJ;HH COOT-
BeTCTByroT <popMYJIaM HCqHCJIeHIDI; HaqaJIbHaH n03HIJ;IDI
39
3. A6cOAIOTHbie AOKa3aTeAbCTBa HenpOTa.1BOpe4a.1BOCTa.1
napTHH (HJIH JII060H maxMaTHoH 3 aqH) cooTBeTCTByeT
Ha60py aKCHOM HCqHCJIeHHH; nOCJIe m:He n03H HH -
<popMYJIaM, BbIBQ HMbIM H3 aKCHOM (T. e. TeopeMaM); Ha-
KOHe , npaBHJIa HrpbI cooTBeTCTB)'IOT npaBHJIaM BbIBO-
a (npaBHJIaM npe06pa30BaHHH) HCqHCJIeHHH. AH8JI0rHH
npocTHpaeTcH H 8JIbme. XOTH caMH no ce6e n03H HH
(pacnOJIO)KeHHH <pHryp f Ha ocKe), no 06Ho <poPMYJIaM
HCqHCJIeHHH, «6ecCMbICJIeHHbI», BbICKa3bIBaHHH 06 STHX
n03H HHX, no 06Ho MeTaMaTeMaTHqeCKHM BbICKa3bIBa-
HHHM 0 <POPMYJIax, BnOJIHe OCMbICJIeHHbI.
«MeTamaxMaTHOe» yrBep eHHe MO)KeT, HanpHMep, rJIa-
CHTb, q'fO B aHHoH n03H HH Y 6e.nbIX B03MO)KHbI Ba-
u
aTb pa3.7IHqHbIX XO OB, HJIH, CKIDKeM, qTO B aHHOH
n03H HH 6e.nbIe, HaqHHaH, MOIyr 3aMaTOBaTb qepHbIX
3a TpH xo a. BOJIee Toro, MO)KHO rOBopHTb H 06 06-
m:HX «MeTamaxMaTHbIX» TeopeMax, B OKa3aTe.nbCTBax
KOTOpbIX HCnOJIb3yeTCH H8JIHqHe JIHmb KOHeqHOrO qHC-
JIa B03MO)KHbIX n03H HH. MO)KHO, HanpHMep, nOJIytlHTb
TeopeMY OTHOCHTe.nbHO qHCJIa B03MO)KHbIX XO OB ,lVI.H 6e-
JIbIX B Haq8JIbHOH (HJIH JII060H pyrOH) n03H HH; HJIH,
CKIDKeM, OKa3aTb TeopeM)', COrJIaCHO KOTOpOH Ba 6e.nbIX
KOHH C KOpOJIeM He MOryr <pOpCHpOBaTb MaT O HHOKO-
MY qepHOMY KOpOJIIO. 9TH H pyrHe «MeTamaxMaTHbIe»
TeopeMbI y aeTcH, TaKHM 06pa30M, OKa3bIBaTb, nOJIb-
3YHCb <pHHHTHbIMH MeTo aMH paCC)'iK,AeHHH, T. e. HCCJIe-
u
A)'H JIHmb KOHeqHOe qHCJIO B03MO)KHbIX n03H HH, y o-
BJIeTBopHIOm:HX qeTKO C<POPMYJIHpOBaHHbIM YCJIOBHHM.
COBepmeHHO aH8JI0rnqHO e.nb rHJIb6epTOBCKOH TeopHH
OKa3aTe.nbCTBa COCTOHT B OKa3aTe.nbCTBe TaKoro )Ke
po a <pHHHTHbIMH MeTo aMH HeB03MO)KHOCTH BbIBo a
npOTHBOpeqam: HX pyr pyry <POPMYJI B aHHOM MaTe-
MaTHqeCKOM HCqHCJIeHHH.
4
CHcreMaTHlieCKoe nocrpoeHHe
OpMaAbHOH AOfHKH
IIpe e qeM rrepeiiTH K caMOH TeOpeMe re eJUI, HaM rrpH-
eTCH rrpeo OJIeTb em:e ABa rrpeIDITCTBWI. npe e Bcero
HaM H8AO pa306paThCH, 3aqeM, co6CTBeHHO, eMY rrOH8Ao6H-
JIaCb Principia Mathematica YaHTXe a H PaCCeJIa H B qeM
Cyrb STOH cHCTeMbI; 8JIee, HaM rroH o6HTCH paCCMOT-
peTb B KaqeCTBe rrpHMepa <popM8JIH3a HH eAYKTHBHOH
CHCTeMbI O HH He6oJIbIIIOH <pparMeHT CHCTeMbI Princip-
ia, H rrOKa3aTb, KaK MO)KHO rro.nyqHTb a6COJIIOTHOe OKa-
3aTeJIbCTBO HerrpOTHBOpeqHBOCTH SToro <pparMeHTa.
06bIqHO, IDKe eCJIH MaTeMaTHqeCKHe AOK 3aTeJIbCTBa
rrpOBOAHTCH C C06JIIO eHHeM o6m:errpHHHTbIX HOpM rrpo-
<peccHOH8JIbHOH CTporOCTH, STa CTporOCTb cym:eCTBeHHO
)'MaJlHeTCH B pe3YJIbTaTe HeKOToporo yrrpom:eHIDI BeCb-
Ma rrpHH HrrH8JIbHOrO xapaKTepa. eJIO B TOM, qTO npHH-
HrrbI (rrpaBHJIa) BhIBo a, yrrOTIJe6.7I.HeMbIe B OKa3aTeJIb-
CTBax, B HBHOH <popMe He <POPMYJIHp)'IOTCH, TaK t.ITO MaTe-
MaTHKH rrpHMeHHIOT HX He BrrOJIHe OC03HaHHO. B03bMeM,
HarrpHMep, eBKJI OBCKOe OKa3aTeJIbCTBO Toro <paKTa,
qTO He cym:eCTBYeT HaH6oJIbIIIerO rrpOCToro qHCJIa ( e-
41
4. CHcTeMaTH....ecKoe nocTpoeHHe <popMaAbHoH AorHKH
JIoe qHCJIO, KaK H3BeCTHo, Ha3bIBaeTCH rrpoCTbIM, eCJIH
OHO He eJIHTCH 6e3 OCTaTKa HH Ha O HO qHCJIO, KpOMe
e HHHnbI H caMoro ce6H). OKa3aTeJIbCTBO, rrpoBo HMoe
MeTO OM reductio ad absurdum (OT rrpoTHBHoro), BbI-
r HT CJIe InHM 06pa30M.
llYCTb, B rrpOTHBOpeqHH C OKa3I>,IBaeMbIM yrBep eHH-
eM, HMeeTCH HaH60JIbmee rrpOCToe qHCJIO. 0603HaqHM
ero qepe3 «x». Tor a:
1. x eCTb HaH60JIbmee rrpOCToe qHCJIO.
2. 06pa3yeM rrpoH3Be eHHe Bcex rrpoCTbIX qHCeJI, MeHb-
mHX HJIH paBHbIx x, H rrpH6aBHM K 3TOMY rrpoH3Be-
eHHIO qHCJIO 1. B pe3YJIbTaTe rrOJIytlHM HeKoTopoe
qHCJIO Y:
Y = (2 x 3 x 5 x 7 x . .. x x) + 1.
3. ECJIH Y caMO eCTb rrpoCToe qHCJIO, TO x He eCTb
HaH60JIbmee rrpOCToe qHCJIO, TaK KaK Y, OqeB HO,
60JIbme x.
4. ECJIH Y - COCTaBHoe qHCJIO (T. e. He HB.7I.HeTCH rrpo-
CTbIM), TO H Tor a x He eCTb HaH60JIbmee rrpOCToe
qHCJIO; B caMOM eJIe, eCJIH Y - COCTaBHoe, TO OHO
u u
OJI)KHO HMeTh HeKOTopbIH rrpOCTOH eJIHTeJIb z;
HO Z HerrpeMeHHO OJI)KHO 6bITb OTJlHqHbIM OT Bcex
rrpOCTbIX qHCeJI 2, 3, 5, 7, . . . , x, MeHbmHX HJIH paB-
HbIX x, TaK qrO Z OJI)KHO B 3TOM CJIyq:ae 6bITb
rrpoCTbIM qHCJIOM, rrpeBOCxOAHInHM x.
5. Ho Y eCTb JIH60 rrpocToe, JIH60 COCTaBHoe qHCJIO.
6. CJIe OBaTeJIbHO, x He eCTb HaH60JIbmee rrpOCToe
qHCJIO.
7. HaH60JIbmerO rrpocToro qHCJIa He CYIneCTByeT.
42
4. CHCTeMaTH....ecKoe nocTpoeHHe c.popMaAbHoH AorHKH
MbI BbIrrHCaJIH 3 ecb TOJIbKO OCHOBHbIe marH OKa3a-
TeJIbCTBa. MO)KHO, O HaKO, rrOKa3aTb, qTO WIH BOCrrOJI-
u u
HeHIDI BceH errOqKH pacc eHHH TaK HJIH HHaqe rrpH-
llIJIOCb 6bI HCrrOJIb30BaTb HeKOTopbIe HeHBHO rro pa3YMe-
BaeMbIe rrpaBHJIa BbIBo a H 3aKOHbI (TeopeMbI) JIOrHKH.
HeKOTopbIe H3 3THX rrpaBHJI H 3aKOHOB rrpHHa.z:vIe)KaT
caMOH 3JIeMeHTapHOH qaCTH <pOpMaJIbHOH JIOrnKH, PY-
rHe - 60JIee BbICOKHM ee pa3 eJIaM, HarrpHMep rrpa-
BHJIa H 3aKoHbI, COCTaBJUlIOInHe TaK Ha3bIBaeM)'1O «Teo-
pHIO KBaJIH<pHKa HH». B STOH TeopHH <POPMYJIHp)'1OTCH
rrpaBHJIa yrrOTpe6JIeHIDI «KBaHTopHbIX» 060pOTOB peqH,
Bpo e «Bce», «HeKOTopbIe» H HX CHHOHHMOB. TIpHBe-
u u
eM 3 eCb rrpHMepbI SJIeMeHTapHOH JIOrHqeCKOH Teo-
peMbI H rrpaBHJIa BbIBo a, HCrrOJIb3yeMbIe, XOTH H He-
HBHO, B rrpHBe eHHOM BbIme OKa3aTeJIbCTBe TeopeMbI
EBKJI a.
06paTHTe BHHMaHHe Ha 5-H mar SToro OKa3aTeJIbCTBa.
OTKY a OH, c06CTBeHHO, rrOJIyqeH? - 113 JIOrHqeCKOH
TeopeMbI (<<He06xo HMOH HCTHHbI»), COrJIaCHO KOTOpOH
«JIH60 p, JIH60 He p», r e qepe3 «p» 0603HaqeHa rrepe-
MeHHaH ( «rrporr03H HOHaJIbHaH rrepeMeHHaH» ). Ho KaK)Ke
HMeHHO 5-H mar OKa3aTeJIbCTBa rro.rryqaeTCH H3 STOR
TeopeMbI? nOcpe CTBOM rrpaBHJIa BbIBo a, Ha3bIBaeMoro
«rrpaBHJIOM rrO CTaHOBKH BMeCTO rrporr03H HOHaJIbHbIX
rrepeMeHHbIX», COrJIaCHO KOTOPOMY H3 JII06oro BbICKa-
3bIBaHIDI MO)KHO BbIBeCTH pyroe BbICKa3bIBaHHe, rro -
CTaBJIIDI BMeCTO KIDK,nOrO BXO)l(,l(eHIDI B HCXO Hoe BbICKa-
u u u
3bIBaHHe HeKOTopOH rrporr03HnHOHaJIbHOH rrepeMeHHOH
(B HameM rrpHMepe rrepeMeHHOH «p») JII060ro (O Horo
H Toro )Ke) BbICKa3bIBaHIDI (B paCCMaTpHBaeMOM CJIyqae
BbICKa3bIBaHIDI «y - rrpOCToe qHCJIO»). TIpHMeHeHHe Ta-
Koro po a rrpaBHJI H JIOrnqeCKHX TeopeM, KaK MbI )')Ke
43
4. CHcTeMaTH4ecKoe nocTpoeHHe <popMaAbHot:1 AorHKH
OTMeqaJIH, IIpOHCXO HT Ha KIDK,nOM ma HO qaCTO co-
BepmeHHo HeOC03HaHHbIM 06pa30M. HBHaH )Ke <POPMY-
JIHpOBKa IIpaBHJI (,r(IDKe ,lVIH CTOJIb IIpoCToro CJIytlaH, KaK
TeopeMa EBKJI a) eCTb OCTH)KeHHe JIHmb 1I0CJIe HerO
CTOJIeTIDI B HCTOpHH JIOrHKH.
nO 06HO MOJIbepOBcKoMY rocllo HY )Kyp eHY, BCIO iKH3Hb
rOBopHBmeMY IIp030H, HO He 1I0 03peBaBmeMY 06 STOM
06CTOHTe.nbcTBe, MaTeMaTHKH B TeqeHHe 110 KpaHHeH Me-
pe BYX TbICHqe.neTHH 06XO HJIHCb 6e3 TOqHOH <pOPMYJIH-
pOBKH IIpHH HIIOB, JIe)KaII(HX B OCHOBe Bcex HX pacC)')K-
eHHH. nOHHMaHHe 1I0AJIHHHOH IIpHpO bI TaKHX IIpHH-
HIIOB - OCTH)KeHHe caMoro He aBHero BpeMeHH.
nOqTH Be TbICHqH JIeT apHCTOTe.neBCKaH TeopIDI IIpa-
BHJIbHbIX <pOpM JIOrHqeCKoro BbIBo a 6e30roBopOqHO CqH-
u u u
TaJIaCb HCqepllbIBalOII(eH H He alOII(eHCH B aJIbHeH-
meH pa3pa6oTKe. EII(e B 1787 r. I1MMaH}'WI KaHT rOBO-
pHJI, qTO <pOpMaJIbH)'lO JIOrHK}' ApHCTOTeJUI «He IIpO BH-
Hemb aJIbme HH Ha O HH IIIar - 3TO HaH60JIee 3aBep-
meHHaH H 1I0JIHaH H3 Bcex HaYK». Ha CaMOM)Ke e.ne Tpa-
H HOHHaH JIOrHKa CyII(eCTBeHHeHIIIHM 06pa30M He 1I0JI-
Ha, H cpe CTB ee He OCTaTOqHO ,lVIH 060CHOBaHIDI MHO-
rHX IIpHH HIIOB BbIBo a, HCIIOJIb3yeMbIx IDKe BO BIIOJIHe
3JIeMeHTapHbIX MaTeMarnqeCKHX pacc eHIDIX.
* npOCTblM npHMepOM MOryr CAY)KHTb npHHUHnbl, HcnoAb3yeMbie npH CAe-
AYlOweM BbIBOAe: 5 > 3, cAeAoBaTeAbHo, 52 > 3 2 .
B03po eHHe JIOrHqeCKHX HCCJIe OBaHHH B HOBoe BpeMH
HaqaJIOCb C olly6JIHKOBaHIDI «MaTeMaTHqeCKOrO aHaJIH3a
JIOrHKH» .l()KopA>Ka EyJUI (1847). EYJIb H ero 1I0CJIe 0-
BaTe.nH 3aHHMaJIHCb IIpe e Bcero pa3pa60TKOH TaK Ha-
3bIBaeMoH aJIre6pbI JIOrHKH, 1I0CBHII(eHHOH BbIHCHeHHIO
44
4. CHCTeMaTH....ecKoe nocTpoeHHe cpopMaAbHoH AOfHKH
H yrOqHeHHIO 60JIee 06 HX H 60JIee pa3HOo6pa3HbIx TH-
v
nOB JIOrHqeCKOH eAYKQHH, He)KeJIH nO nMalOInHe no
TpMH HoHHbIe JIOrHqeCKHe npHH HnbI. C nOMOInblO 6y-
v
JIeBOH TeXHHKH JIerKO BbIpa)KaIOTCH, KOHeqHO, H TpMH-
HoHHbIe )'M03aKJIlOqeHIDI.
pyroe HanpaBJIeHHe HCCJIe OBaHHH, TeCHO CBH3aHHoe
C pa3pa6oTKOH MaTeMaTHKaMH XIX CTOJIeTIDI npo6JIe-
MaTHKH oCHoBaHHH aH8JIH3a, TaIOKe OKa38JIOCb 6JIH3KHM
nporpaMMe EyJUl. QeJIblO HOBoro HanpaBJIeHIDI 6bIJIO
npe CTaBHTb BCIO qHCTyIO MaTeMaTHK)' KaK qaCTb cpop-
M8JIbHOH JIOrHKH. KnacCHqeCKOe BbIpIDKeHHe 3Ta JIHHIDI
pa3BHTIDI JIOrHKH H MaTeMaTHKH nOJIyt-lHJIa B Princip-
ia Mathematica YaHTXe a H PacceJIa (1910-1913). Ma-
TeMaTHKaM XIX-ro CTOJIeTIDI y 8JIOCb «apHcpMeTH3Hpo-
BaTb» 8JIre6py H TaK Ha3bIBaeMoe «HCqHCJIeHHe 6ec-
KOHeqHO M8JIbIX», nOKa3aB, qTO pa3JIHqHbIe nOIDITIDI,
HCnOJIb3yeMbIe B MaTeMaTHqeCKOM aH8JIH3e, Onpe eJIH-
MbI HCKJIlOqHTeJIbHO B apHcpMeTHqeCKHX TepMHHax (T. e.
B TepMHHax eJIbIX qHCeJI H apHcpMeTHqeCKHX onepa HH
HM HHMH). HanpHMep, BMeCTO Toro qTo6bI onycKaTb
MHHMoe qHCJIO R B KaqeCTBe HeKoeH MHCTHqeCKOH
«C HOCTH», ero CT8JIH Onpe eJI.HTb KaK yrroPMOqeH-
H)'IO napy eJIbIX qHCe.n (0, 1), npHqeM HM TaKHMH napa-
MH pa3pemeHo 6bIJIO npoH3Bo HTb onpe eJIeHHOrO po a
onepa HH «CJIO)KeHIDI» H «)'MHO)KeHIDI». .AH8JIOrHqHO,
Hppa HOH8JIbHOe qHCJIO v'2 Tenepb CT8JIH Onpe eJI.HTb
v
KaK HeKoTopbIH KJIacc pa HOH8JIbHbIX qHCeJI, a HMeH-
HO, KaK KJIacc pa HOH8JIbHbIX q.HCeJI, KBMpaTbI KOTOpbIX
MeHbme 2. PacceJI )Ke (a eIne paHee HeMe KHH MaTeMa-
THK rOTTJIo6 <l>pere) nOCTaBHJI CBoeH eJIblO nOKa3aTb,
qTO Bce apU!pMemU1.LeCKUe nOHRmWl MO)KHO onpe e.nHTb
B qHCTO JIOrHqeCKHX TepMHHax, a Bce aKCHOMbI apHcp-
45
4. CHcTeMaTH....ecKoe nocTpoeHHe <popMaAbHot1 AorHKH
MeTHKH BbIBeCTH H3 He60JIbIIIOrO qHCJIa rrpe,lVIO)KeHHH,
KOTopbIe MO)KHO 6bIJIO 6bI KB8JIH<pHQHpOBaTb KaK qHCTO
JIOrHqeCKHe HCTHHbI.
npHBe eM rrpHMep. B JIOrHKe HMeeTCH rrOHHTHe K/lacca.
,[{Ba KJIacca, rro orrpe e.neHHIO, «rro 06HbI», eCJIH Me
HX qneHaMH MO)KHO YCTaHoBHTb B3aHMHo-0 H03HaqHOe
COOTBeTCTBHe (rrpHqeM rrOHH';rHe B3aHMHo-0 H03HaqHO-
ro COOTBeTCTBIDI caMO MO)KeT 6bITb orrpe e.neHo B Tep-
MHHax pyrHx JIOrHqeCKHX rroHHTHH). Knacc, HMelOInHH
u
e HHcTBeHHbIH qneH, Ha3bIBaeTCH «e HHHqHbIM KJIac-
COM» (TaKOB, HarrpHMep, KJIaCC eCTeCTBeHHbIX crryrHH-
KOB 3eMJIH); Kap HH8JIbHOe (KOJIHqeCTBeHHoe) qHCJIO 1
orrpe eJIHeTCH KaK KJIacc Bcex KJIaCCOB, rro 06HbIX KaKO-
MY-JIH60 e HHHqHOMY KJIaccy. AH8JI0rnqHO MO)KHO orrpe-
eJIHTb H pyrHe Kap HH8JIbHbIe qHCJIa; pa3JIWlHbIe apH<p-
MeTHqeCKHe orrepaQHH (CJIO)KeHHe, YMHO)KeHHe H T. .)
TaIOKe MO)KHO orrpe e.nHTb qepe3 rrOHHTIDI <pOpM8JIbHOH
JIOrHKH. npOH3BOJIbHOe apH<pMeTHqeCKOe yrBep eHHe
. (cKa)KeM, «1 + 1 = 2») MO)KHO Terrepb rrpe CTaBHTb
KaK COKpaIneHH)'IO 3arrHCb HeKoToporo yrBep eHIDI, co-
u
CTaBJIeHHOrO HCKJIlOqHTe.nbHO H3 BbIpa)KeHHH, rrpHHM-
JIe)KaInHX 06bIqHOH JIOrHKe, H Bce TaKHe qHCTO JIOrH-
qeCKHe yrBep eHIDI, KaK MO)KHO rrOKa3aTb, BbIBO HMbI
u
H3 HeKoTopoH CHCTeMbI JIOrHqeCKHX aKCHOM.
TaKHM 06pa30M, Principia Mathematica HBHJIaCb c e-
CTBeHHbIM rrpO BH)KeHHeM B peIIIeHHH rrp06JIeMbI He-
rrpOTHBOpeqHBOCTH MaTeMaTHqeCKHX CHCTeM, B qaCTHO-
CTH apH<pMeTHKH, B TOM CMbICJIe, qTO rrocpe CTBOM STOH
CHCTeMbI R M. 6bIJIO OCTHrHyrO HeKOTopoe CBe eHHe
yrrOMHHyrOH rrp06JIeMbI K rrp06JIeMe HerrpOTHBOpeqHBO-
CTH caMOH <pOpM8JIbHOH JIOrHKH. B caMOM e.ne, eCJIH aK-
46
4. CHCTeMaTH....ecKoe nocTpoeHHe cpopMaAbHoH AorHKH
CHOMbI apH<pMeTHKH Cyrb npocTo-HanpocTo cOKpa eH-
HbIe 3anHCH HeKoTopbIX TeopeM JIOrHKH, TO BOnpOC 0 TOM,
COBMeCTHMbI JIH apH<pMemqeCKHe aKCHOMbI, 3KBHBaJIeH-
TeH Bonpocy 0 COBMeCTHMOCTH OCHOBHbIX JIOrHqeCKHX
aKCHOM.
,[{aJIeKO He Bce MaTeMaTHKH (no pa3HbIM npHqHHaM) CO-
rJIaCRJIHCb C Te3HCOM <l>pere- PaCCeJIa, COrJIaCHO KOTOpO-
MY MaTeMaTHKa eCTb He qTO HHoe, KaK qaCTb JIOrHKH.
KpOMe Toro, KaK MbI )')Ke OTMeqaJIH, aHTHHOMHH KaHTO-
pOBCKOH TeopHH 6eCKOHeqHbIX MHO)KeCTB, eCJIH He npH-
HHTb cneQHaJIbHbIX Mep npe OCTOpO)KHOCTH, JIerKO BOC-
npOH3BO,WITCH H B paMKax qHCTOH JIOrHKH. Ho He3aBH-
CHMO OT CTeneHH npHeMJIeMOCTH caMoro no ce6e Te3Hca
<l>pere-PaCCeJIa Ba OCTOHHCTBa CHCTeMbI R M. n03BO-
JUlIOT CqHTaTb ee HeOQeHHMbIM OCTH)KeHHeM Ha nyrH
K aJIbHeHIIIeMY H3yqeHHIO np06JIeMbI HenpOTHBOpeqH-
BOCTH. B Principia pa3pa60TaHa 3aMeqaTeJIbHa.H cBoeH
KpaTKOCTblO CHCTeMa 0603HaqeHHH, npH nOMO H KOTO-
pOH Bce npeAJIO)KeHIDI qHCTOH MaTeMaTHKH (B qaCTHO-
CTH, apH<pMeTHKH) MOIyr 6bITb 3anHcaHbI HeKOTopbIM
CTaHAapTHbIM 06pa30M. KpoMe Toro, B 3TOH KHHre HB-
HbIM 06pa30M C<POPMYJIHpOBaHo 60JIbIIIHHCTBO npaBHJI
BbIBo a, HCnOJIb3yeMbIx B MaTeMaTHqeCKHX OKa3aTeJIb-
CTBax (6bITb MO)KeT, H3BeCTHbIX H paHee, HO He B CTOJIb
TOqHOM H nOJIHOM BH e). Pe3IOMHPYH, MO)KHO CKa3aTb,
qTO B Principia C03 aH BeCbMa cOBepIIIeHHbIH HHCTpY-
MeHT ,lVI.H HCCJIe OBaHIDI BceH CHCTeMbI apH<pMeTHKH KaK
HeHHTepnpempoBaHHoro HCqHCJIeHIDI, T. e. KaK CHCTe-
MbI 6eCCMbICJIeHHbIX 3HaqKOB, H3 KOTOpbIX nocpe crBoM
TOqHO C<pOPMYJIHpOBaHHbIx npaBRJI 06pa3yIDTCH H npe-
06pa3yIDTCH «CTpOqKH» 3HaKOB - <POPMYJIbI.
5
OAHH npHMep
a6COAIOTHOrO AOKa3aTeAbCTBa
HenpOTHBOpe HBOCTH
HaM IIp eTCH Tenepb BbIIIOJIHHTb BTOpyID 3a,z:(aqy H3 yrro-
MHHJTbIX B Haqane npe M ero pa3 eJIa H 03HaKO-
MHTbCH C O HHM BIDKHbIM, XOTH H BIIOJIHe OcryrrHbIM,
IIpHMepoM a6COJIIOTHOrO OKa3aTeJIbCTBa HenpOTHBope-
qHBOCTH. YCBOHB 3TO OKa3aTeJIbCTBO, qHTaTeJIb CMO)KeT
JIytlIIIe o eHHTb 3HaqeHHe pa60TbI re eJI.H.
MbI nOKIDKeM 3 ecb KOpOTKO, KaK MO)KHO <popManH-
u
30BaTb 3JIeMeHTapHYID JIOrHK)' BbIcKa3bIBaHHH, HBJlHIO-
CH HeKOTopbIM <pparMeHTOM cHcTeMbI, onHcaHHOH
B Principia Mathematica. B pe3YJIbTaTe <popManH3a HH
yrrOMHHYTbIH <pparMeHT Principia CTaHeT HCqHCJIeHHeM,
COCTOHID;HM H3 HeHHTenpeTHpOBaHHbIX CHMBOJIOB. nOCJIe
3Toro MbI )"Ke CMO)KeM IIpoBe TH H)"KHee HaM OKa3a-
TeJIbCTBO.
<l>opManH3a IDI IIpOXO HT B qeTbIpe 3Tana. npe)K]l:e Bce-
ro HaM IIOH o6HTCH IIOJIHbIH IIepeqeHb CHMBOJIOB, KO-
TopbIe HCIIOJIb3yIDTCH B HaIIIeM HCqHCJIeHHH, OHH CO-
49
5. 0.6J1H npHMep a6coAIOTHoro .6.0Ka3aTeAbCTBa HenpoTHBope4HBocTH
CTaBHT TaK Ha3bIBaeMbIH aJI<l>aBHT CHCTeMbI. aJIee HaM
HMO 6YAeT C<t>OpM)'JIHpOBaTb «npaBHJIa 06pa30BaHIDI»,
COrJIaCHO KOTOpbIM H3 «6YKB» aJI<t>aBHTa COCTaBJIHIOT-
CH «<t>OPMYJIbI» (npHqeM TOJIbKO TaKHe, «npaBHJIbHO CO-
CTaBJIeHHbIe», COqeTaHIDI CHMBOJIOB MbI 6YAeM CqliTaTb
npe,lVIO>KeHIDIMH HameH CHCTeMbI). MO>KHO 6bIJIO 6bI
u
CqHTaTb COBOKynHOCTb npaBHJI «rpaMMaTHKOH» HCqHC-
JIeHIDI. 3aTeM MbI oroHpaeM HeKOTopbIe <t>OPMYJIbI Ha-
meR CHCTeMbI B KaqeCTBe ee aKCHOM (HJIH «HCXOAHbIX
<t>OPMYJI»), aKCHOMbI CJI)')KaT «6a3HcoM» CHCTeMbI. II,
HaKOHeQ, MbI C<t>OpM)'JIHpyeM «npaBHJIa npeo6pa30Ba-
HIDI», TOqHO onHCbIBalO He, KaKHM o6pa30M H3 OAHHX
<t>OPMYJI HeKOToporo BHAa «BbIBO,wITCH» APyrHe <t>OPMY-
JIbIOnpeAeneHHOrOBHAa;HHaqerOBOpH,npaBHJIaSTH-
He qTO HHoe, KaK npaBHJIa BbIBOAa. Teope.M.ou. HameR
CHCTeMbI MbI 6YAeM Ha3bIBaTb Tenepb JII06yIO <t>OPMY.ny,
no.nyqaeMyro nocpeAcTBoM nOCJIeAOBaTenbHoro npH e-
HeHIDI npaBHJI npeo6pa30BaHIDI K aKCHOMaM. <l>OpMaJIb-
HbIM «AoKa3aTenbcTBoM» MbI 6YAeM Ha3bIBaTb JII06yro
KOHeqH)'lO nOCJIeAOBaTenbHocTb <t>OPMYJI paccMaTpHBae-
Moro HCqHCJIeHIDI, K aH H3 KOTOpbIX JIH60 HBJIHeTCH
aKcHoMoR, JIH60 BbIBoAHMa H3 npeAmeCTByro Hx <t>op-
u
M)'JI AaHHoH nOCJIeAOBaTenbHocTH C nOMo blO npaBHJI
npeo6pa30BaHIDI 1 ) .
A.n<t>aBHT JIOrHKH BbIcKa3bIBaHHR (Ha3bIBaeMoR qaCTO
«npono3HQHOHaJIbHbIM HCqHCJIeHHeM») OqeHb HeCJIO>KeH.
OH . COCTOHT H3 nepeMeHHbIx H KOHCTaHT. IIepeMeHHbIe,
1) 113 3Toro onpe e.neHHH HeMe;:vIeHHO BbiTeKaeT, qTO aKCHOMbl TaK-
)Ke npHqHCJIHIOTCH K TeopeMaM OKa3aTe.nbCTBO Ka OH TaKOH Teope-
Mbl COCTOHT H3 e HHCTBeHHOH (j?OPMYJIbl - H3 Hee caMOH). - IIpUM.
nepe8.
50
5. 0.6J1H npHMep a6coAIOTHoro .6.0Ka3aTeAbCTBa HenpoTHBope4HBocTH
nOCKOJIbK)' BMeCTO HHX MO>KHO nOACTaBJUITb npe,lVIo>Ke-
HHH (sentences) cHcTeMbI, Ha3bIBalOT CeHTeHIJ;HOHaJIb-
HbIMH (qa e - npOn03HIJ;HOHaJIbHbIMH) nepeMeHHbIMH.
B KaqeCTBe nepeMeHHbIx MbI 6YAeM HCnOJIb30BaTb 6YKBbI
«p», «q», «r», ... , «PI », «P2» ... , «ql », «q2» ... ·
IIOCTOHHHbIe CHMBOJIbI (KOHCTaHTbI) - STO «npon03H-
IJ;HOHaJIbHbIe» CBH3KH H 3HaKH npenHHaHHH. MbI 6YAeM
ynOTpe6J1HTb CJIeAYID He npOn03HIJ;HOHaJIbHbIe CBH3KH:
«f'V» qHTaeTCH KaK «He»; V - «HJIH»; «::>>> - «eCJIH...,
TO... »; «.» - «H»; 3HaKH npenHHaHHH: «(» - «JIeBaH
cK06Ka», «)>> - «npaBaH cK06Ka».
AeHCTBHTeAbHO, nepe4HcAeHHbie CBSl3KH B03HHKAH KaK cOKpaweHHble 060- +
3Ha4eHHSI MSI YKa3aHHbix B cK06Kax Bblpa)KeHHH; 60Aee Toro, npH YCTHOM
4TeHHH cpOPMYA HC4HCAeHHSI BblCKa3blBaHHH 3THMH Bblpa)KeHHSlMH 4aCTO
Ha3blBalOT cooTBeTcTBYlOwHe cpopMaAbHble CHMBOAbl (cK(l)KeM, cpopMYAa
«f'J p V q» 4HTaeTCSI KaK «He p HAH q» 1.1 T. n.). CAe.6.yeT, O.6.HaKO, TBep.6.0
nOMHHTb, 4TO 3TH «Ha3BaHHSI» CBSl30K He H}')KHbl MSI onHcaHHSI HC4HCAeHHSI
(HeHHTepnpeTHpOBaHHoro!) KaK TaKOBoro; OHH OTHOCSlTC5I K ero MeTaTeo-
pl1l1, 1.1, CKa)KeM, 3AeKTpOHHO-BbI4HCAHTeAbHa51 MaWHHa, npOH3BO.6.5IWa51
onepaUHH C cpopMYAaMH HC4HCAeHHSI BblCKa3blBaHHH KaK C Ta KOBbiM11/
B TaKoro pO.6.a «Ha3BaHHSlX» He HY)KAaeTCSI. - npl1M. nepeB.
IIpaBHJIa 06pa30BaHHH YKa3bIBaIOT, KaKHe HMeHHO KOM-
6HHaIJ;HH SJIeMeHTapHbIX CHMBOJIOB aJI<paBHTa MbI 6YAeM
CqHTaTb <popMyJIaMH Hamero HCqHCJIeHHH. IIpe e Bce-
ro <POPMYJIOH, no onpeAeJIeHHIO, HBJlHeTCH KIDKAaH npo-
n03HIJ;HOHaJIbHaH nepeMeHHaH. aJIee, eCJIH «8» 060-
3HaqaeT HeKOTOpyro <POPMY.ny ), TO ee «<pOpMaJIbHOe OT-
pHIJ;aHHe» «f'V (8)>> TalOKe eCTb <popMyJIa. AHaJIOrHq-
2) HMeHHO 0603Ha'lClem, HO He Sl.B/lSlemCSl (j?OPMYJIOH (HBJUleTCH
UMeHe.M (j?0PMYJIbl); S, He npHHa,wIe)Ka aH a.n(j?aBHTY onHCblBaeMoro
HCQHCJleHHH, OTHOCHTCH K ero MemWl3bLKy. - IIpUM. nepeB.
51
s. 0.6J1H npHMep a6COAIOTHOrO .6.0Ka3aTeAbCTBa HenpOTHBOpe4HBOCTH
HO, eCJIH «8 1 » H «8 2 » Cyrb 0603HaqeHIDI HeKoTopbIX
<POPMYJI, TO BbIpIDKeHIDI «(8 1 ) V (8 2 ) », «(8 1 ) ::> (8 2 )>>
H «(8 1 ) · (8 2 )>> TaIOKe Cyrb <P0PMYJIbI.
IIpHMepbI <poPMYJI:
«p», «rv p», «(P)::> (q)>>, «((q) V (r)) ::> (P)>>.
Q,l\HaKO BbIpa>KeHIDI «(P)(rv q)>> HJIH «((P) ::> (q))V» <pOp-
MYJIaMH He HBJIHIOTCH, TaK KaK OHH He Y,l\OBJIeTBOpHIOT
npHBe,l\eHHOMY 3AeCb onpe,l\eJIeHHIO <pOPMYJIbI 3) .
IIpaBRJI npeo6pa30BaHHH HMeeTCH ,l\Ba. IIepBoe H3 HHX -
npaBWlO nOacmaHOBKU (BMeCTO npOn03HD:HOHaJIbHbIX
nepeMeHHbIX) - rJIaCHT, QTO H3 npOH3BOJIbHOH <pOPMYJIbt
MO>KHO BbIBeCTH ,l\PYJYIO <POPMYJIY nOCpe,l\CTBOM O,l\HO-
BpeMeHHOH IIOACTaHOBKH HeKOTopOH <pOPMYJIbI BMeCTO
HeKOTopOH BXO,wIrneH B HCXO,l\HYIO <POPMYJIY npon03H-
u u
D:HOHaJIbHOH nepeMeHHOH, npHQeM TaKaH nO,l\CTaHOBKa
(O,l\Ha H Ta >Ke) ,l\OJI>KHa npOH3BO,l\HTbCH BMeCTO Ka:Hc-
a020 BXO:»caeHWl BbI6paHHOH nepeMeHHOH. HanpHMep,
H3 <POPMYJIbI «p ::> p» MO>KHO, nO,l\CTaBHB BMeCTO nepe-
MeHHOH «p» nepeMeHHYIO (a TeM caMbIM - <P0PMYJIY)
«q », BbIBeCTH <POPMYJIY «q ::> q»; nO,l\CTaBHB B TY >Ke
HCXO,l\HYIO <POPMYJIY BMeCTO «p» <POPMYJIY «p V q», MbI
BbIBe,l\eM <POPMYJIY «(p V q) ::> (p V q)>> H T. n. MJIH, eCJIH
HHTepnpeTHpCBaTb «p» H «q» KaK HeKOTopbIe PYCCKHe
npeAJIO>KeHIDI, TO H3 «p ::> p» MO>KHO, HanpHMep, nOJIY-
QHTb npeAJIO>KeHIDI «JlHryIIIKH KBaKalOT ::> JIHryIIIKH
KBaKaIOT», «(JIerryqHe MbIlliH CJIenbI V JIeTytIHe MbIlliH
e,wIT MbImeH) ::> (JIerryqHe MbImH CJIenbI V JIerryqHe MbI-
mH e,WIT MbImeH)>> H T. n. BTopoe npaBHJIO npe06pa-
3) B Tex c.nyqamc, Kor a HeqerO onacaTbCH He opa3YMeHHH, qaCTb
CKo60K B 3anHCHX (j?OPMYJIbl onycKalOT.
52
5. OAHH npHMep a6coAIOTHoro AOKa3aTeAbCTBa HenpoTHBope4HBocTH
30BaHlliI - STO TaK Ha3bIBaeMoe npasWlo omaen.eHWl
(RJIH modus ponens). COrJIaCHO STOMY npaBHJIY H3 JIIO-
6bIX ABYX <poPMYJI, HMelO HX COOTBeTCTBeHHO BHA «8 1 »
H «8 1 ::> 8 2 », MO>KHO BbIBecrH H <poPMYJIY «8 2 ». HanpH-
Mep, H3 <POPMYJI «pV f"V p» H «(pV f"V p) ::> (p ::> p)>> MbI
MO>KeM BbIBeCTH «p ::> p».
HaKoHeu;, aKCHOMaMH Hamero HCqHCJIeHIDI (no c e-
CTBY TeMH >Ke, qTO B Principia Mathematica 4) HBJIHIOTCH
CJIeAYID He qeTbIpe <pOPMYJIbI 5):
1. (PVp)::>p
[eCJIH p RJIH p, TO p];
2. P ::> (p '! q)
[eCJIH p, TO P HJIH q];
3. (p V q) ::> (q V p)
[eCJIH p HJIH q, TO q R7IH p];
4. (p::> q) ::> ((r V p) ::> (r V q))
[eCJIH p BJIeqeT q, TO (r HJIH p) BJIeqeT (r RJIH q )].
3AeCb BHaqane npHBeAeHbI aKcHoMbI, a B KBa,z:(paTHbIx
cKo6Kax yKa3aHbI HX «nepeBoAbI» Ha 06bIqHbIH H3bIK 6 ).
4) B Principia 6bL7Ia e e aKCHOMa «(p V (g V r)) V (g V (p V r))»
BbIBO HMaH, O HaKO, KaK YCTaHOBHJI II. BepHaHc (1926), H3 OCTaJIbHblX
qeTblpex aKCHOM. - IIpUM. nepea.
5) HaqHHaH OTCIO a, Mbl 6y eM, KaK 06blqHO, onycKaTb KaBblqKH
npH 3anHCHX (j?opM)']I, HaneqaTaHHblX B OT e.nbHYJO CTpOK)'. HaM, Be b,
H)?KHbl He caMH no ce6e KaBblqKH, a'YBepeHHOCTb B TOM, qTO He B03-
HHKHeT He opa3YMeHHH (cp. C Ha3BaHHeM KHHrH Pacce.na H YaHTXe a,
BCIO B HaCTOH eH KHH)I(Ke Bb emIeMblM He KaBblqKaMH, a lCYpCU-
aOM). - IIpUM. nepea.
6) «IIepeBo bl» 3TH, pa3YMeeTCH, K caMOM)' HCqHCJIeHHIO He OTHO-
CHTCH. - IIpUM. nepea.
53
5. OAHH npHMep a6coAIOTHoro AOKa3aTeAbCTBa HenpoTHBope4HBocTH
K aH H3 npHBeAeHHbIx aKCHOM npeACTaB.IDIeTCH AO-
u u
BOJIbHO-TaKH «OqeBHAHOH» H TpHBHaJIbHOH.
+ ECAH, KOHe4HO, HMeTb B BHAY HeKoropble «eCTeCTBeHHble nepeBoAbl» (T. e.
HHTepnpeTauHH!) aKCHOM, caMHX no ce6e HHKaKoro «CMbICAa» He HMelO-
WHX. AHaAorH4Hoe 3aMe4aHHe cAeAye-r HMeTb B BHAY npH 4TeHHH CAeAY-
IOwei1 <ppa3bl TeKCTa H BCIOAY B aHaAOrH4HbiX CA}"-IaSiX AaAee. - npl4M.
nepeB.
TeM He MeHee H3 HHX C nOMO bIO C<pOPMYJIHpOBaH-
HbIX BbIllie ABYX npaBHJI npeo6pa30BaHHH MO)KHO BbIBe-
CTH 6eCKOHeqHOe MHO)KeCTBO TeopeM, MHorne H3 KOTO-
pbIX TpYAHO Ha3BaTb OqeBHAHbIMH RJIH TpHBHaJIbHbIMH.
K qHc.rry TaKHX TeopeM OTHOCHTCH, CKIDKeM, <poPMYJIa
((P ::> q) ::> ((r ::> 8) ::> t)) ::>
::> ((u ::> ((r ::> 8) ::> t)) ::> ((P ::> u) ::> (8 ::> t))).
B AaHHbIH MOMeHT Hac, OAHaKO, He HHTepecyeT BbIBOA
TeopeM H3 aKCHOM. QeJIb Hallia COCTOHT B TOM, qrr06bI
u
nOKa3aTb HenpOTHBOpeqHBOCTb STOH CHCTeMbI aKCHOM,
T. e. AaTb «a6COJIIOTHoe» AOKa3aTeJIbCTBO Hea03MO)/Q-lO-
emu BbIBOAa H3 AaHHbIX aKCHOM C nOMO blO npaBRJI
npeo6pa30BaHHH HHKaKOH <POPMYJI@ S OAHOBpeMeHHO
C ee <pOpMaJIbHbIM OTpHQaHHeM rvS.
OKa3bIBaeTCH, qTO K qHc.rry TeopeM Hamero HCqHCJIe-
HHH OTHOCHTCH <poPMYJIa «p ::> (rv p ::> q)>> (BbIpIDKae-
MaH CJIOBeCHO CJIeAYIO HM 06pa30M: «eCJIH p, TO He p
BJIeqeT q»). (MbI npHMeM STOT pe3YJIbTaT K cBeAeHHIO,
He npOBO,wI <paKTHqeCKOrO ero AOKa3aTeJIbCTBa.) L(ony-
CTHM, qTO HeKOTOpaH <popMYJIa S, TaK )Ke KaK H ee OTpH-
QaHHe rv S, BbIBOAHMa H3 aKCHOM. IIOACTaBJIHH TorAa S
54
5. OAHH npHMep a6coAIOTHoro AOKa3aTeAbCTBa HenpoTHBope4HBocTH
u
BMeCTO nepeMeHHoH «p» B TOJIbKO qTO ynOMHHYfYIO Teo-
peM)' (nOJIb3YHCb npaBRJIOM nOACTaHoBKH) H npHMeHHH
3aTeM AB bI modus ponens, MbI no.nyqHM, qTO Teope-
MOH HB.1llIeTCH H <popMYJIa «q».
nOACTaBAS1S1 S BMeCTO (p) B «P ::> (f'J P ::> q)>>, Mbl nOAY4HM CHa4aAa +
«S ::> (f'J S ::> q) ». 6epS1 3aTeM 3TY cpOPMYAY 1.1 cpOPMYAY S B Ka4eCTBe
nOCblAOK modus ponens, nOAY4HM «f'J S f'J q». HaKOHeu, 1.13 nocAeAHetli
cpOPMYAbl 1.1 f'J S TalOKe no modus ponens nOAY4HM cpOPMYAY «q».
Ho eCJIH <popMYJIa, COCTOH aH H3 oAHoH-eAHHCTBeHHOH
u u
nepeMeHHoH «q», HB.1llIeTCH TeopeMOH, TO nOCKOJIbK)'
BMeCTO «q» MO>KHO nOACTaBHTb luo6yro lfiopMYIlY, TO
1l106Wl lfiopMYlla Hawe20 UC1.LUCIleHWl OKa3bLBaemC.R BbL-
BOiJuMou U3 aKCUOM. OTCIOAa BHAHO, qTO eCJIH KaKaH-
JIH60 <popM)'JIa S BMeCTe CO CBOHM OTpHQaHHeM rv S
u
HB.1llIeTCH TeopeMOH paCCMaTpHBaeMoro HCqHCJIeHIDI, TO
B HeM TeopeMOH HB.1llIeTCH JII06aH <popMYJIa. Kopoqe ro-
BOpH, K aH <popMYJIa npOTHBOpeqHBOrO HCqHCJIeHIDI
u u
HB.1llIeTCH TeopeMOH - H3 npOTHBOpeqHBOH CHCTeMbI aK-
CHOM MO>KHO BbIBecm JII06yro <POPMY.ny: Ho STOT >Ke pe-
3YJIbTaT MO)KHO BbIpa3HTb H B «06paTHOH» <popMe: eCJIH
He K aH <popM)'JIa HCqHCJIeHIDI HB.1llIeTCH TeopeMOH
(T. e. HMeeTCH XOTH 6bI OAHa <pOPMYJIa, He BbIBOAHMaH
H3 AaHHbIX aKCHOM), TO STO HCqHCJIeHHe HenpOTHBope-
qHBO. TaKUM 06Pa30M, Hawa 3aaa1.la CBOaUmC.R K mOMY,
1.lm06bL nOKa3amb, 1.lmo UMeemC.R no KpauHeU.Mepe OaHa
lfiop.MYlla, He BbLBOaUMWl U3 paCCMampUBae.M.OU cucme-
MbL aKCUOM.
3a,naqa MO>KeT 6bITb pemeHa nocpeACTBOM HeKOToporo
MeTaMaTeMaTHqeCKOrO pacc eHIDI 0 paCCMaTpHBae-
MOH CHCTeMe. llAeH TaKoro pacc eHIDI BeCbMa npo-
3paqHa. Cyrb ee CBOAHTCH K Haxo eHHIO HeKOToporo
55
s. O.6.HH npHMep a6COAIOTHOrO .6.0Ka3aTeAbCTBa HenpOTHBOpe4HBOCTH
CTpYKTypHoro CBOHCTBa <POPMYJI AaHHOH CHCTeMbI, YAo-
BJIeTBOpHlOrnerO CJIeA)'lOrnHM TpeM YCJIOBIDIM:
(1) CBOHCTBO STO AOJDKHO BbInOJIHHTbCH ,lVIH Bcex qe-
TbIpex aKCHOM.
(2) CBOHCTBO STO AOJI>KHO 6bITb «HaCJIeACTBeHHbIM»
no OTHOllIeHHIO K npaBHJIaM npe06pa30BaHIDI; HHa-
qe rOBopH, eCJIH OHO npHcyrne BceM aKCHOMaM, TO
OHO AOJI>KHO npHHaw:re>KaTb H JII060H <popMyJIe, BbI-
BOAHMOH H3 STHX aKCHOM. A nOCKOJIbKy <popMYJIa,
BbIBOAHMaH H3 aKCHOM, eCTb, no onpeAeJIeHHIO, Teo-
peMa, TO AaHHoe YCJIOBHe CBOAHTCH K TOM)) qrrO HC-
KOMbIM CBOHCTBOM AOJDKHa 06JIa,naTb K aH Teo-
peMa.
(3) HCKOMOMy CBOHCTBY AOJI>KHbI YAOBJIeTBOpHTb He Bce
<POPMYJIbI, KOTopbIe MO>KHO nOCTpOHTb C nOMOrnbIO
npaBHJI 06pa30BaHIDI AaHHOH CHCTeMbI. MbI AOJDK-
u u
HbI )'MeTb nOKa3aTb, qrrO no KpaHHeH Mepe OAHa
<popMYJIa CHCTeMbI STHM CBOHCTBOM He 06JIa,naeT.
ECJIH HaM YAaCTCH HaHTH CBOHCTBO <pOpMyJI CHCTeMbI,
YAOBJIeTBOpHIOrnee nepeqHCJIeHHbIM TpeM YCJIOBIDIM, TO
3a,naqa nOCTpoeHIDI a6COJIIOTHOrO AOKa3aTeJIbCTBa He-
npOTHBOpeqHBOCTH CHCTeMbI 6YAeT pellieHa. B caMOM Ae-
JIe, STO CBOHCTBO, 6Y H HaCJIeACTBeHHbIM H npHHaw:re-
>Ka aKCHOMaM, npHHaw:re>KHT H TeopeMaM; 3HaqHT, eCJIH
HeKOTopoe 3HaKOCOqeTaHHe, HBJlHHCb <POPMYJIOH AaHHOH
CHCTeMbI, He 06JIa,naeT YKa3aHHbIM CBOHCTBOM, TO STO -
He TeopeMa. HHaqe rOBopH, eCJIH qneH, nOA03peBaeMbIH
B npHHaw:re>KHOCTH HeKoeMY ceMeHCTBY (<popMYJIa), JIH-
llIeH <paMHJIbHbIX qepT, npHcyrnHx K OMY HaCTOHrneMy
qneH)' ceMeHCTBa (HAYrnHX OT 06rnHX npeAKOB - aKCH-
OM), TO OH Ha caMOM AeJIe He MO>KeT npHHaw:re>KaTb STO-
56
s. OAHH npHMep a6COAIOTHOrO AOKa3aTeAbCTBa HenpOTHBOpe4HBOCTH
ro KJIallY (6bITb TeOpeMOH). Ho eCJIH HaM YAaJIOCb HaHTH
<POPMY.ny AaHHOH CHCTeMbI, He HB.1llIIO CH TeopeMOH,
u
TO MbI TeM caMbIM AOKa3aJIH HenpOTHBOpeqHBOCTb STOH
CHCTeMbI - BeAb, KaK MbI COBceM HeAaBHO OTMeqaJIH,
u u
B CHCTeMe, He HB.1llIIO eHCH HenpOTHBOpeqHBOH, Ka aH
<poPMYJIa BbIBOAHMa H3 aKCHOM (T. e. K aH <popMYJIa
HB.JIHeTCH TeopeMOH). Kopoqe rOBopH, Bce, qTO HaM Ha-
u u
AO ,lVIH perneHHH HarneH 3a,naqH, - STO HaHTH XOTb OAIIY
<POPMYJI)) He o6JIa,nalO HaCJIeACTBeHHbIM CBOHCTBOM,
YAOBJIeTBOpHIO HM onHcaHHbIM BbIrne YCJIOBHHM.
B KaqeCTBe TaKoro CBOHCTBa rOAHTCH, HanpHMep, CBOH-
CTBO «6bITb TaBTOJIOrHeH». BbI 3HaeTe, qTO TaK 06bIqHO
HMeH)'lOT yrBep eHIDI, AB bI nOBTopHIO He BHernHe
pa3JIHqHbIM 06pa30M OAIIY H 'IY >Ke MbICJIb H He Hecy-
He nOSTOMY <paKTHqeCKH HHKaKOH HH<popMaQHH. Ha-
npHMep, «pa3 L(>KOH eCTb OTeQ QapJI3a, TO QapJI3 - CbIH
L(>KOHa». B o606 eHHe SToro CBoHcTBa «HeHH<popMa-
THBHocm» B JIOrHKe TaBTOJIOrIDIMH npHIDITo Ha3bIBaTb
yrBep eHHH, KOTopbIe He MOryr He 6bITb HCTHHHbI-
MH. TIpHMepoM MO>KeT CJI)')KHTb BbICKa3bIBaHHe: «AO)K]J:b
HAeT HJIH AO b He HAeT». rOBOpHT TalOKe, qTO TaBTO-
JIOrHH - «HCTHHbI BO Bcex B03MO>KHbIX MHpax», R7IH,
e e nO-APyroMY, qTO STO Heo6xoaUMO (R7IH /102U1.LecKU)
ucmUHHbLe BbICKa3bIBaHHH.
Ho,lVIH Toro tfr06bI Harne AOKa3aTeJIbCTBO HenpOTHBope-
qHBOCTH 6bIJIO He OTHOCHTeJIbHbIM, a a6COJIIOTHbIM, HaM
np eTCH AaTb TaKoe onpeAeJIe.HHe nOIDITHH TaBTOJIOrHH,
KOTopoe He 3aBHCeJIO 6bI HenOCpeACTBeHHO OT nOIDITHH
HCTHHbI (B CBOIO OqepeAb, nOApa3YMeBalO ero HeKOTO-
pyro uHmepnpema14UlO), a 6bIJIO 6bI AaHO B qHCTO <pop-
MaJIbHbIX, CTpYKTYPHbIX TepMHHax.
57
s. OAHH npHMep a6COAIOTHOrO AOKa3aTeAbCTBa HenpOTHBOpe4HBOCTH
HanOMHHM, tfrO <popMyJIa Hamero HCqHCJIeHIDI - JIH-
60 npoCTO OAHa H3 6YKB, HCnOJIb3yeMbIx B HeM B Ka-
qeCTBe npon03HD;HOHaJIbHbIX nepeMeHHbIX CHa30BeM Ta-
KHe <POPMYJIbI «SJIeMeHTapHbIMH»), ]lH60 )Ke COCTaBJIeHa
H3 TaKHX 6)1KB C nOMoW;bIO npon03HD;HOHaJIbHbIX CBH30K
H cK060K. YCJIOBHMCH OTHeCTH K SJIeMeHTapHYIO
<POPMYJIY B OAHH H3 ABYX HenepeceKalOW;HxcH KJIaCCOB,
B cYMMe AalOW;Hx Bce MHO)KeCTBO <pOpMyJI HCqHCJIeHIDI -
K 1 RJIH K 2 . <l>OPMYJIbI, He HBmlIOW;HeCH SJIeMeHTapHbI-
MH, OTHOCHTCH K TOMY RJIH HHOMy H3 STHX KJIaCCOB B CH.1IY
u
CJIeAYIOW;HX COrJIameHHH:
1) <popMYJIa, HMelOIIJ;aH BIIA 8 1 V 8 2 , npHHa.n;Ie)KHT
KJIaccy K 2 , eCJIH KaK 8 1 , TaK H 8 2 npHHa.n;Ie)KaT
K 2; B npOTHBHOM c.nyqae OHa npHHa.n;Ie)KHT K 1 ;
2) <popMYJIa, HMelOW;aH BIIA 8 1 :) 8 2 , npHHa.n;Ie)KHT
KJIaccy K 2 , eCJIH 8 1 npHHa.n;Ie)KHT K 1 , a 8 2 npH-
Ha.n;Ie)KHT K 2; B npOTHBHOM c.nyqae OHa npHH e-
)KHT K 1 ;
3) <popMyJIa, HMelOW;aH B 8 1 .8 2 , npHH8)I,7Ie>KHT KJIacey
K 1 , eCJIH KaK 8 1 , TaK H 8 2 npHHa.n;Ie)KaT K 1 ; B npo-
THBHOM c.nyqae OHa npHHa.n;Ie)KHT K 2 ;
4) <popMYJIa, HMelOW;aH BIIA f"V 8, npHHa.n;Ie)KHT KJIaccy
K 2, eCJIH 8 npHHa.n;Ie)KHT K 1; B npOTHBHOM c.nyqae
OHa npHHa.n;Ie)KHT K 1 .
Tenepb MbI onpeAeJIHeM CBORCTBO «6bITb TaBTOJIOrHeR»:
<popMYJIa eCTb TaBTOJIOrIDI TorAa H TOJIbKO TorAa, KorAa
OHa npHHa.n;Ie)KHT KJIaccy K 1 He3aBHCHMO OT Toro, Ka-
KOMY H3 KJIaCCOB K 1 H K 2 npHHa.n;Ie)KHT JII06aH H3 BXO-
AHW;HX B Hee SJIeMeHTapHbIX <POPMYJI CT. e. nepeMeHHbIX).
RCHO, tfrO STO onpeAMeHHe He HCnOJIb3yeT HHKaKOR MO-
AMH HJIH HHTepnpeTaD;HH HameR CHCTeMbI. MbI MO)KeM
58
5. O.6.HH npHMep a6coAIOTHoro .6.0Ka3aTeAbCTBa HenpoTHBope4HBocTH
YCTaHOBHTb, HBJIHeTCH JIH KaKaH-JIH6o AaHHaH cpoPMY-
u
JIa TaBTOJIOrHeH, npoCTo HCCJIeAy-H ee cTpoeHHe C TOq-
KH 3peHHH BbInOJIHeHHH npHBeAeHHbIx BbIIlle qeTbIpex
u
YCJIOBHH.
TaKaH npoBepKa npHBoAHT K BbIBOA)', tlTO KIDKAaH H3 qe-
TbIpex aKCHOM HBJIHeTCH TaBTOJIOrHeM. TIpoQe a TaKoM
npoBepKH CBOAHTCH K COCTaBJIeHHIO Ta6JIHQbI, B KOTOpOM
yqHTbIBalOTcH Bce B03MO)KHbIe BapHaHTbI COOTHeceHHH
SJIeMeHTapHbIx KOMnOHeHT AaHHoM aKCHOMbI K JII060MY
H3 ABYX KJIaCCOB, Kl H K 2 . TIpocMaTpHBaH nOCJIeAOBa-
TeJIbHO CTpOKH TaKoM Ta6JIHQbI, MbI MO)KeM OnpeAeJIHTb
u
,lVIH KIDKAorO H3 B03MO)KHbIX pacnpeAeJIeHHH «3Haqe-
HHM» (T. e. npHHClAJIe)KHOCTH KJIaCCaM Kl H K 2 ) SJIe-
MeHTapHbIx cpoPMYJI (T. e. nonpocry nepeMeHHbIx), KaKO-
MY H3 KJIaCCOB npHHClAJIe)KHT KIDKAaH HeSJIeMeHTapHaH
«nOAcpOpMYJIa» AaHHOM cpOPMYJIbI H BCH paCCMaTpHBa-
eMaH cpOpMYJIa B QeJIOM. B03bMeM, HanpHMep, nepByro
aKCHOMY. Ta6JIHQa ,lVIH Hee COCTOHT H3 Tpex CTOJI6QOB:
u u
nepBbIH H3 HHX COOTBeTCTByeT eAHHCTBeHHOH ee SJIe-
u u
MeHTapHOH KOMnOHeHTe «p», BTOpOH - HeSJIeMeHTap-
HOM nOAcpOpMYJIe «(p V p)>>, a TpeTHM - BceM cpopMYJIe
«(p V p) :) p». B KIDKAOM H3 CTOJI6QOB YKa3aHbI KJIac-
CbI, KOTOpbIM npHHClAJIe)KaT COOTBeTCTByro He CPOPMY-
u
JIbI npH AaHHbIX pacnpeAeJIeHIDIX 3HaqeHHH nepeMeH-
HbIX no STHM KJIaCcaM. BOT KaK BbIr.1l.HAHT Ta6JIHQa ,lVIH
u
nepBoH aKCHOMbI:
p p V P (p V p) :) P
Kl Kl Kl
K 2 K 2 Kl
59
5. OAHH npHMep a6COl\IOTHOrO AOKa3aTel\bCTBa HenpoTHBope4HBocTH
B nepBoM cToJI6ne Ta6JIHnhI npHBe,neHhI B03MO)KHhIe
v v
3HaqeHIDI e,nHHCTBeHHOH 3JIeMeHTapHOH KOMnOHeHTbI pac-
v
cMaTpHBaeMoH aKcHoMhI, BO BTOpOM - cOOTBeTCTByroInHe
3HaqeHIDI He3JIeMeHTapHoH KOMnOHeHThI aKCHOMhI (co-
rJIaCHO YCJIOBHIO (1), B TpeTheM - 3HaqeHIDI caMOH aK-
CHOMhI (cOrJIaCHO YCJIOBHIO (2)). 113 nOCJIe,nHerO CTOJI6na
Cpa3Y BH,nHO, qTO nepBaH aKCHOMa npHHa,nJIe)KHT KJIaccy
K 1 BCer,na, He3aBHCHMO OT TOrO, K KaKOMY KJIaccy OT-
HeCeHa ee 3JIeMeHTapHaH KOMnOHeHTa. 3 HaqHT, nepBaH
v
aKCHOMa HBJUleTCH TaBTOJIOmeH.
A BOT TaKaH )Ke Ta6JIHna ,n.T.UI BTOpOH aKCHOMhI:
p q pVq P -:J (p V q)
Kl Kl Kl Kl
Kl K 2 Kl Kl
K 2 Kl Kl Kl
K 2 K 2 K 2 Kl
B nepBhIX ,nBYX CTOJI6nax Ta6JIHnhI yKa3aHhI Bce B03MO)K-
HhIe pacnpe,ne.neHIDI ,nBYX 3JIeMeHTapHhIX KOMnOHeHT aK-
CHOMhI no ,nB)'M KJIaCCaM, B TpeTheM - COOTBeTCTByroInHe
3HaqeHIDI ee He3JIeMeHTapHOH KOMnOHeHThI (COrJIaCHO
YCJIOBHIO (1)), B qeTBepToM - 3HaqeHIDI caMoH aKCH-
OMhI. 11 3,neCh H3 paccMoTpeHIDI nOCJIe,nHero cToJI6na
Ta6JIHnhI cpa3Y B HO, qTO aKCHOMa HBJUleTCH TaBTOJIO-
meH. TOqHO TaK )Ke YCTaHaBJIHBaeTCH TaBTOJIOrHqHOCTh
OCTaJIhHhIX ,nBYX aKCHOM.
,l(oKIDKeM Teneph, qTO CBOHCTBO «6hITh TaBTOJIOmeH» Ha-
CJIe,nCTBeHHO OTHOCHTeJIhHO npHMeHeHHH npaBHJIa modus
60
s. OAHH npHMep a6COl\IOTHOrO AOKa3aTel\bCTBa HenpOTHBOpe4HBOCTH
ponens. (.l(oKa3aTeJIhCTBO ero HaCJIe,nCTBeHHOCTH OTHO-
CHTeJIhHO npaBHJIa no,nCTaHOBKH npe,nOCTaBJUleTCH tlH-
TaTeJIlO.) IIYCTh <POPMYJIhI 8 1 H 8 1 :) 8 2 - TaBTOJIOrHH;
HaM H O ,nOKa3aTh, tlTO Tor,na H <popMYJIa 8 2 eCTh TaB-
TOJIOrIDI. ,L{onYCTHM, tITO 8 2 He HBJUlerCH TaBTOJIOmeH.
B TaKOM CJIyqae ,z:v:uI XOTH 6hI o,nHoro pacnpe,neJIeHIDI
3JIeMeHTapHhIX KOMnOHeHT 3TOH <pOPMYJIhI no KJIaCCaM
K 1 H K 2 OHa npHH8A7Ie)KHT KJIaccy K 2 . Ho, no npe,n-
nOJIO)KeHHIO, 8 1 HBmIeTCH TaBTOJIOmeH, T. e. npHH -
JIe)KHT KJIaccy K 1 npH JII06hIX pacnpe,neJIeHIDIX CBOHX
3JIeMeHTapHhIX KOMnOHeHT, B TOM tlHCJIe H npH TOM, npH
KOTOpOM 8 2 npHH8A7Ie)KHT K 2 7). Ho Tor,na npH 3TOM
pacnpe,neJIeHHH <popMYJIa 8 1 :) 8 2 ,nOJI)KHa (B CHJIY BTO-
poro YCJIOBIDI) npHH8A7Ie)KaTh KJIaccy K 2 , tlTO, o,nHa-
KO, npoTHBOpetlHT npe,nnOJIO)KeHHIO 0 TaBTOJIOrHtlHOCTH
8 1 :) 8 2 . IIpoTHBopetlHe nOKa3hIBaeT, tlTO 8 2 ,nOJI)KHa
6hITh TaBTOJIOrHeH. TaKHM 06pa30M, TaBTOJIOmtlHOCTh
cpOPMYJIhI eCTh CBOHCTBO HaCJIe,nCTBeHHOe, T. e. nepe,na-
BaeMoe OT nOChlJIOK npaBHJIa modus ponens K ero 3a-
KJIlOtleHHIO.
Teneph HaM OCTaerCH yKa3aTh npHMep <pOPMYJIhI Hame-
ro HCtlHCJIeHIDI, He HBJUlIOIneHCH TaBTOJIOrHeH. TaKOBa,
HanpHMep, <poPMYJIa «p V q», npHH8A7Ie)KaInaH KJIaccy
K 2, eCJIH o6e ee KOMnOHeHThI ( «p» H «q») npHH8A7Ie)KaT
3TOMY KJIaccy 8). (B nepeBo,ne Ha co,nep)KaTeJIhHhIH H3hIK:
BhICKa3hIBaHHe «" p" HJIH q"» JIO:>KHO, eCJIH JIO:>KHhI o6a
BXO,lUlInHe B ero COCTaB BhICK 3hIBaHIDI «p» H «q».)
7} IlpWieM CKa3aHHoe BepHO 6e3OTHOCHTeJIbHO K TOM)', BXO HT JIH
B <pOpM)'JIbI 8 1 H 82 XOTb O,lUla o6 aB nepeMeHHaB. - IIpUM. nepea.
8} KOHeQHO, e e 60JIee npoCToH npHMep - <popM)'JIa, COCTOH aB
H3 o HoH-e HHCTBeHHoH nepeMeHHoH p. - IIpUM. nepea.
61
5. OAHH npHMep a6COl\IOTHOrO AOKa3aTel\bCTBa HenpoTHBope4HBocTH
HaIIIa QeJIh AOCTHrHYTa. Mhl HaIIIJIH <POPMY.ny; He HBJUI-
IO CH TeopeMOH HameH CHCTeMhl. Ho B cnyqae npoTH-
BOpetlHBOCTH Bhl6paHHoH HaMH CHCTeMhl aKCHOM TaKoH
<POPMYJIhl B HaIIIeM HCtlHCJIeHHH He HaIIIJIOCh 6hl. TaKHM
06pa30M, H3 aKCHOM HCtlHCJIeHHH BhlCKa3hlBaHHH HeJIh3H
BhlBeCTH HHKaKOH <pOpMyJIhl o,nHOBpeMeHHO C ee OTpHQa-
HHeM. 9THM H 3aBepIIIaeTCH a6COJIIOTHOe ,nOKa3aTeJIhCTBO
HenpOTHBOpetlHBOCTH HCtlHCJIeHHH BhICKa3hIBaHHH.
JIerKo BH,neTh, tlTO KJIaCChl Kl H K 2 MO)KHO nOHHMaTh
COOTBeTCTBeHHO KaK KJIaCC ucmUHHbLX H KJIaCC ILO)/c-
HbLX BhICKa3bIBaHHH. Mhl, o,nHaKO, HaMepeHHO B03,nep-
v
jJ(HBaJIHCh OT 3TOH TepMHHOJIOrHH B xo,ne caMoro ,no-
Ka3aTeJIhCTBa (XOTH He pa3, KOMMeHmupYR OT,neJIhHhle
ee IIIarH, no,npa3)'MeBaJIH B03MO>KHOCTh ee HCnOJIh30Ba-
HHH), tlT06hl no,ntlepKHYTh TO 06CTOHTeJIhCTBO, tlTO HaIIIe
,nOKa3aTeJIhCTBO B npHH Hne He aeTCH B CChIJIKax
Ha KaK)'lO 6hl TO HH 6hlJI0 HHTepnperaQHIO <POPMYJI HC-
v .
tlHCJIeHHH BhICKa3hIBaHHH, XOTH nOHHTh ero KaK CJIe,nyeT
JIeNe HMeHHO npH TaKOM «nepeBo,ne» Ha co,nep>KaTeJIh-
HhlH H3hIK.
B 3aKJIIOtleHHe CJIe,nyeT CKa3aTh eIne 06 O,nHOH BIDK-
HOH np06JIeMe, oTHoCHIll;eHcH K HCtlHCJIeHHIO BhlCKa3hI-
BaHHH. MhI YCTaHOBHJIH, tlTO K aH TeopeMa 3Toro HC-
v
tlHCJIeHHH HBJIHerCH TaBTOJIOrHeH, T. e. - eCJIH BhIpa-
>KaThCH B TepMHHax Heo,nHoKpaTHo ynoMHHaeMoH BhlIIIe
v v
co,nep>KaTeJIhHOH HHTepnpeTa HH - JIOrHtleCKOH HCTH-
v
HOH, «3aKOHOM JIOrHKH». ECTeCTBeHHO 3MaTb B H3BeCT-
HOH Mepe H 06paTHhlH Bonpoc: K oe JIH JIOrHtleCKH
HCTHHHoe BhICKa3hIBaHHe, BhIpa3HMoe Ha H3hlKe HaIIIe-
ro HCtlHCJIeHIDI (T. e. K aH JIH TaBTOJIOrHH), HBJUlerCH
TeopeMoH ,naHHoro HCtlHCJIeHIDI (BhIBO,nHMOH H3 ero aK-
62
5. OAI1H npl1Mep a6COl\IOTHOrO AOKa3aTel\bCTBa HenpOTI1BOpe411BOCTI1
CHOM)? M Ha 3TOT Bonpoc MO)KHO ,naTh nOJIO>KHTeJIhHhIH
OTBeT; HO ,nOKa3aTeJIhCTBO TaKoro <t>aKTa CJIHIlIKOM ,lVIHH-
HO, tIT06hI npHBo,nHTb ero 3,neCh. Ho HaM XOTeJIOCh 6hI
06paTHTh BHHMaHHe Ha O,nHO 06CTOHTeJIhCTBO, He HMelO-
ee OTHOllieHIDI K caMOMY ,nOKa3aTeJIhCTBY: ,neJIO B TOM,
tlTO pe3YJIhTaT 3TOT CBH,nereJIhCTByer 0 ,nOCTaTOtlHOCTH
BhI6paHHhIX HaMH aKCHOM ,lVI.H nOJIyqeHIDI BCex TaBTO-
JIOrHtlHhIX <pOpM)'JI - HHhIMH CJIOBaMH, BCex JIOrHtleCKH
u
HCTHHHhIX BhICKa3hIBaHHH, BhIpa3HMhIX Ha H3hIKe HCtlHC-
JIeHIDI BhICKa3hIBaHHH. CHCTeMhI aKCHOM, 06JIMalO He
u
TaKHM CBOHCTBOM, npHIDITO Ha3hIBaTh «nOJIHhIMH».
Bonpoc 0 nOJIHOTe TOH H.JIH HHOH CHCTeMhI aKCHOM npe,n-
CTaBJI.Her, KaK npaBH.JIO, 60JIhllIOH HHTepec. B caMOM ,ne-
JIe, OCHOBHhIM CTHM)'JIOM ,lVI.H aKCHOMaTH3anHH pa3JIHtI-
HhIX pa3,ne.TIOB MaTeMaTHKH 6hIBaer cTpeMJIeHHe HaHTH
u u
no,nxO,lUl HH nepetleHh HCXO,nHhIX ,nonym:eHHH, H3 KO-
TOphIX 3aTeM MO)KHO 6hmo 6hI BhIBeCTH Bce HCTHHHhIe
npe,lVIO)KeHIDI ,naHHoH 06JIaCTH. CKIDKeM, Kor,na EBKJIH,n
<t>OpM)'JIHpOBan HeKoTopYlO aKCHOMaTH3anHIO 3JIeMeH-
TapHoH reoMeTpHH, OH CTapancH oTo6paTh aKCHOMhI Ta-
KHM o6pa30M, tIT06hI H3 HHX MO)KHO 6hmo BhIBeCTH
Bce HCTHHHhIe reoMerplNeCKHe yrBep eHIDI, He TOJIh-
KO )')Ke H3BeCTHhIe B TO BpeMH, HO B npHHnHne H JII06hIe
,npyrHe, KOTophIe MO)KHO 6hmo 6hI HayqHThcH ,nOKa3bI-
BaTh Kor,na-JIH60 B 6Y,nyIneM.
nOMI1MO npo4ero, EBKAI1A 06Hap)')K111\ nopa311Tel\bHYIO npOHI1UaTel\bHOCTb +
cBoeH TpaKTOBKOH 3HaMeHI1TOH aKCI10Mbi napal\l\el\bHOCTI1 KaK Aonywe-
HI1S1, I\Orl14eCKI1 He 3aBI1CSlWerO OT OCTal\bHblX aKCI10M npeAAO)l(eHHOH 11M
CI1CTeMbl. I\I1Wb cnyCTSI MHoro BpeMeHI1 YAal\OCb AOKa3aTb, 4TO 3Ta aK-
CI10Ma AeHCTBI1Tel\bHO He MO)l(eT 6blTb BblBeAeHa 113 OCTal\bHblX aKCJ.10M
EBKAJ.1Aa, T. e. 4TO 6e3 a KCJ.10M bl napaMel\bHOCTI1 3Ta CJ.1CTeMa aKCJ.10M
HenOI\Ha.
63
5. OAHH npHMep a6COl\IOTHOrO AOKa3aTel\bCTBa HenpoTHBope4HBocTH
O He,naBHero BpeMeHH <1t1HTaJIOCb 60JIee HJIH MeHee ca-
MO C060H pa3yMeIOInHMC.H, tlTO AJUI KIDK,AOH KOHKpeTHoH
06JIaCTH MaTeMaTHKH MO>KHO no,no6paTb nOJIH)'IO CHCTe-
MY aKCHOM. B tlaCTHOCTH, MaTeMaTHKH 6hIJIH y6em.ne-
HbI, tlTO CHCTeMa aKCHOM, npe,n.nO>KeHH8..H AJUI aKCHOMa-
TH3aQHH apH<pMeTHKH HaT)'paJIhHhIX tlHCeJI, nOJIHa HJIH
BO BC.HKOM CJIyqae MO>KeT 6hITh nOnOJIHeHa (c,neJIaHa nOJI-
HOH) ,no6aBJIeHHeM K HCXO,nHOMY nepetlHIO eIne KOHetl-
Horo cnlfCKa aKCHOM. O,nHHM H3 BeJIHtlaHmHX OTKpbITHH
re,neJUI If 6bIJIO KaK pa3 06Hap)')KeHHe HeB03MO>KHOCTH
TaKoH nOJIHOH aKCHOMaTH3aQHH apH<pMeTHKH.
6
HAe.H KOAHpOBaHH.H
H ee HCnOAb30BaHHe
B MaTeMaTHKe
MCtlHCJIeHHe BhICKa3hIBaHHH npe,nCTaBJUleT C060H npH-
u
Mep MaTeMaTHtleCKOH CHCTeMhI, no OTHomeHHIO K KO-
TOpOH 3MatlH, BhI,nBHraeMhIe rHJIh6epTOBCKOH TeopH-
u
eH ,nO Ka3aTeJIhCTBa, OKa3aJIHCh, KaK MhI BH,neJIH, nOJI-
HOCThIO peaJIH30BaHHhIMH. KOHetlHO, 3TO HCtlHCJIeHHe
<pOpMaJIH3yeT JIHllIh HeKOTophIH <pparMeHT <pOpMaJIh-
u u
HOH JIOrHKH, H3hIK H ,ne HBHhIH annapaT KOTOpO-
ro He,nOCTaTOtlHhI ,nIDKe ,lVI.H <pOpMaJIhHOrO nOCTpoeHIDI
3JIeMeHTapHOH apH<pMeTHKH. Ho B03MO)KHOCTH rHJIh6ep-
u
TOBCKOH nporpaMMhI OTHIO,nh He orpaHINHBalOTCH ,nOKa-
'"
3aTeJIhCTBOM HenpOTHBOpetlHBOCTH HeqHe1IeHlliI BhICKa-
3hIBaHHH. MO)KHO npHBeCTH npHMephI ropa3,no 60JIee
60raThIX TeopHH, ,lVI.H KOTOphIX OKa3aJIOCh B03MO)KHhIM
,naTh CTporo MeTaMaTeMaTHtleCKHe ,nOKa3aTeJIhCTBa He-
npOTHBOpeQHBOCTH H nOJIHOThI. IIpHMepoM MO>KeT c.ny-
>KHTh apHcpMeTlNeCKaH CHCTeMa C onepa HeH C/lo:»ceH.Wl
(HO 6e3 onepa HH YMHO>KeHIDI) HaTYPaJIhHhIX tlHCeJI,
65
6. V1AeSi KOAHpoBaHHSI H ee HCnOl\b30BaHHe B MaTeMaTHKe
AIDI KOTOpOH TaIOKe MO)KHO npoBecTH a6COJIIOTHOe }J.o-
Ka3aTeJIbCTBO HenpOTHBOpetlHBOcTH. Ho }J.ocTaTOtlHO JIH
CHJIbHbI <pHHHTHbIe MeTO}J.hI I'wIb6epTa AIDI YCTaHOBJIe-
HIDI HenpoTHBOpetlHBOCTH CHCTeM BpO}J.e Principia - CH-
creM, BbIpa3HTeJIbHbIe H JIOrHtleCKHe cpe}J.CTBa KOTOpbIX
n03BOJIHIOT nOCTpOHTb BCIO apH<pMeTH a He TOJIhKO
OT}J.eJIhHhIe ee <pparMeHTbI? HeO}J.HOKpaTHbIe nonbITKH
u
HaHTH TaKoe }J.OKa3aTeJIbCTBO ycnexoM He YBeHtlaJIHCh,
a pa60Ta re}J.e.mI 1931 L nOKa3aJIa, tlTO OHH H He Mor-
JIH 6bITh ycnemHhIMH, TaK KaK, CTporo npH}J.ep)KHBaJlCb
rpaHH[(, npe}J.HatlepTaHHbIX B HCXO}J.HOH <pOPMYJIHpOB-
Ke rHJIb6epTOBCKOH nporpaMMbI, 31)' 3Maqy Boo6 e pe-
mHTb HeJIb3H.
qTO )Ke, co6CTBeHHO, }J.OKa3aJI re}J.eJIb H KaK HMeHHO }J.O-
Ka3aJI? B pa60Te re}J.e.mI HMelOTCH }J.Ba OCHOBHbIX pe-
3YJIbTaTa. TIpe)K,lJ;e Bcero (MbI 3}J.eCb He HMeeM B BHJl.Y TOT
u
nOPMOK, B KaKOM 3TH pe3YJIbTaTbI H3JIaralOTCH B caMOH
pa60Te re}J.e.mI) OH }J.OKa3bIBaeT HeB03MO)KHOCTh Meia-
MaTeMaTHtleCKOrO }J.OKa3aTeJIbCTBa HenpOTHBOpetlHBOCTH
. JII060H CHCTeMbI, }J.OCTaTOtlHO 06mHpHOH, tlTo6hI BKJIIO-
tlaTb B ce6H BCIO apH<pMeTHK)', KOTopoe (}J.OKa3aTeJIbCTBO)
He HCnOJIb30BaJIO 6bI KaKHX-JIH60 CyI.I(eCTBeHHO HHhIX
npaBHJI BbIBO}J.a, KpOMe Tex, tlTO HCnOJIb3yIOTCH AIDI BbI-
BO}J.a TeopeM B caMOH paCCMaTpHBaeMOH CHCTeMe. Ko-
HetlHO, H TaKoe (nOJIb3yIOIneeCH 60JIee CHJIbHhIMH B He-
KOTOpOM CMbICJIe npaBHJIaMH BbIBO}J.a) }J.OKa3aTeJIbCTBO
MO)KeT 6bITb OtleHb BIDKHbIM H nOJIe3HbIM. Ho Bce )Ke
eCJIH }J.OKa3aTeJIbCTBO CTpOHTCH Ha OCHOBe npaBHJI BhIBO-
}J.a, 3HatlHTeJIbHO 60JIee MOInHhIX, He)KeJIH JIOrHtleCKHe
cpe}J.CTBa apH<pMerntleCKOrO HCtlHCJIeHIDI, TaK tlTO YBe-
peHHOCTb B HenpOTHBOpetlHBOCTH HCnOJIh3yeMhIx B }J.O-
Ka3aTeJIbCTBe }J.onyI.I(eHHH 6Y}J.eT HHqyTh He 60JIhme, tleM
66
6. V1Ae51 KOAlllpoBaHIllSi III ee IIICnOl\b30BaHille B MaTeMaTIIIKe
paCqeThI Ha HenpoTHBOpeqHBOCTh apH<pMeTHKH, TO u:eH-
HOCTh TaKOrO ,nOKa3aTeJIhCTBa 6y,neT ,nOBOJIhHO-TaKH cne-
nH<pHQeCKOH: MhI y6heM O,nHO tIY,nOBHIne neHOH P0)K-
,neHIDI ,npyroro. Bo BC.HKOM CJIyqae, eCJIH 3TO ,nOKa3a-
TeJIhCTBO 6y,neT He <pHHHTHCTCKHM, TO OCHOBHOH nyHKT
rH.JIh6epTOBCKOH nporpaMMhI OCTaHeTC.H, KOHeqHO, HeBhI-
nOJIHeHHhIM. re,neJIeBCKOe paCCY'KAeHHe KaK pa3 H no-
Ka3hIBaeT BCIO 6eCnOqBeHHOCTh paCqeTOB Ha HaxO)K,l];e-
HHe <pHHHTHCTCKOro ,nOKa3aTeJIhCTBa HenpOTHBOpeqHBO-
CTH apH<pMeTHKH.
BTOpOH OCHOBHOH pe3YJIhTaT pa60ThI re,nemI, nO)KanyH,
eIne 60JIee HeO)lGf,l\aH H nOpa3HTe.1IeH; OH )'Ka3hIBaer Ha He-
KOTOpyro npHHnHllHaJIhHyro orpaHINeHHOCTb B03MO)KHOC-
TeH aKCHOMamqecKoro MeTO,na. re,neJIh nOKa3hIBaer, qTO
CHCTeMa Principia Mathematica, KaK H BC.HKaH HHaH CH-
CTeMa, cpe,nCTBaMH KOTOpOH MO)KHO nOCTpOHTh apH<pMe-
THK)', - cytqecmBeHHO HenO/lHa. 9To 3HaqHT, qTO AJI.H
JII060H ,naH oH He,poTHBOpeqHBOH CHCTeMhI apH<pMe-
THqeCKHX aKcOOM1IMeIOTC.H HCTHHHhIe apH<pMernqeCKHe
u
npeAJIO)KeHIDI, He BhIBo,nHMhIe H3 aKCHOM 3TOH CHCTeMhI.
9TO 06CTO.HTeJIhCTBO Hrpaer pemalOIn)'IO pOJIh,nJI.H ou:eH-
KH BceH pa60TbI re,neJUl, H Ha HeM CTOHT OCTaHOBHThC.H
HeCKOJIhKO no,npo6Hee. MaTeMaTHKaM xopomo H3BeCT-
HhI npHMephI 06InHX ymepiK](eHHH, ,nJI.H KOTOphIX ,no CHX
nop He HaH,neHO HHKaKoro onpOBepralOInerO npHMepa,
HO He HaH,neHo H ,nOKa3aTeJIhCTBa. KnacCHqeCKHM npH-
MepOM TaKoro po,na MO)KeT CJIy)KHTh 3HaMeHHTaH «Teo-
peMa rOJIh,n6axa», yrBep)K,l];aIOInaH, qTO KaiK](Oe qerHOe
qHCJIO MO)KHO npe,ncTaBHTh B BH,ne C)'MMhI ,nBYX npocThIX
qHCeJI. MhI He MO)KeM )'Ka3aTb HH o,nHoro qeTHOrO qHCJIa,
KOTopoe He .HBlliIJIOCh 6hI C)'MMOH ,nBYX npOCThIX, HO y Hac
67
6. lt1AeSi KOAHpoBaHHSI H ee HCnOl\b30BaHHe B MaTeMaTHKe
HeT if ,nOKa3aTeJIhCTBa rHnOTe3hI rOJIh,n6axa, npHro,nHoro
WIH Bcex qeTHhIX qHCeJI. TaKHM o6pa30M, nepe,n HaMH -
apH<pMeTHqeCKOe )'TBep eHHe, KOTopoe BnOJIHe MOiKeT
6hITh HCTHHHhIM, HO He BhIBO,nHMhIM H3 aKCHOM apH<p-
MeTHKH. OnyCTHM, qTO 3TO ,neHCTBHTeJIhHO TaK (yrBep-
aTh no,no6Hoe MhI, pa3)'MeeTCH, He MOiKeM). IIpe,ncTa-
BHM ce6e, qTO MhI H3MeHHJIH HJIH nOnOJIHHJIH HCXO,n-
H)'IO aKCHOMaTHK)' TaKHM 06pa30M, qTo6hI Bce HCTHH-
u
HhIe, HO He BhIBo,nHMhIe B HCXO,nHOH CHCTeMe npe,lVIOiKe-
HIDI (K HX qHCJIy OTHOCHTCH, no c,neJIaHHOMY TOJIhKO qTO
npe,nnOJIOiKeHHIO, rHnOTe3a rOJIh,n6axa) CTaH)'T B paClliH-
peHHOH CHCTeMe BhIBO,nHMhIMH 1). TeopeMa re,neJlH nOKa-
3hIBaeT, qTO HHKaKoe TaKoe paClliHpeHHe apH<pMeTHqe-
u u
CKOH CHCTeMhI He MOiKeT c,neJIaTh ee nOJIHOH" T. e. qTO
,naiKe eMH nOnOJIHHTh ee 6eCKOHeqHhIM MHOiKeCTBOM aK-
u u
CHOM, Bce paBHO B HOBOH CHCTeMe HaH,nyTCH HCTHHHhIe,
HO He BhIBo,nHMhIe (XOTH H BhIpa3HMhIe!) ee cpe,nCTBaMH
npe,lVIOiKeHIDI.
-0. lt1CTHHHOCTb TaKHX npeAAO)f(eHHH, KaK Mbl HH)f(e YBHAHM, MO)f(HO ycra-
HOBHTb nocpeACTBOM HeKOToporo MeTaMaTeMaTH4eCKoro pacc)')KAeHHSI
06 apHcpMeTH4ecKoH CHCTeMe. Ho TaKoe paCCY)l(AeHHe He YAOBl\eTBOpSieT
Tpe6oBaHHIO, COrl\aCHO KOTOPOMY HC4HCI\eHHe AOI\)f(HO 6blTb, TaK CKa3aTb,
«3aMKHYTOH CHCTeMOH», T. e. Bce AOKa3yeMbie B HeM HCT HH npeAAO-
)f(eHHSI AOI\)f(Hbl 6blTb nOl\Y4eHbi KaK cpopMal\bHble Cl\eACTBHSI H3 aKCHOM
BHYTpJ1 caMoro HC4HCl\eHHSI. TaKHM 06pa30M, aKCHOMaTH4eCK H MeTOA
KaK cpeACTBO nOCTpoeHHSI BceH COAep)f(aTel\bHOH apHcpMeTHKH OKa3b1Ba-
eTCSI npl1HUl1nl1a/\bHO orpaHH4eHHblM.
1) TaKoe paCillHpeHHe MO:>KHO npOH3BeCTH, npOCTO npHcoe.n;HHHB
3TH He.n;OKa3yeMbIe npe,lJ;JlO)KeHHH K apHcpMeTHKe B KalleCTBe HOBbIX
UKCUOM.. IlOCKOJIbKY MbI CtlHTaeM HX HCTHHHbIMH, TO OTpHn;aHHH HX
He .n;OJI)KHbI H He MOryr 6bITb .n;OKa3yeMbI B apHcpMeTHKe; 3HallHT, Ta-
Koe paCillHpeHHe HenpOTHBOpellHBOH CHCTeMbI He MO)KeT c.n;eJIaTb ee
npOTHBOpellHBOH. - IIpUM. nepea.
68
6. V1AeSi KOAl1pOBaHI1S1 11 ee I1CnOAb30BaHI1e B MaTeMaTI1Ke
QTo6hI tlHTaTe.TI1O 6hmo JIerqe nOIDITh H,l(elO ,l(OKa3aTe.TIh-
CTBa re,l(eJIH, MhI ( CJIe,l(yB re,l(e.TIIO) npHBe,l(eM BHatlaJIe
cxeMY pacc eHIDI, nOCpe,l(CTBOM KOToporo no.rryqaer-
CH JIOrHtleCKaH aHTHHOMIDI (npoTHBOpetlHe), H3BeCTHaH
no,l( Ha3BaHHeM «napMoKca PHmapa» (no HMeHH onH-
caBmero ee B 1905 r. <ppaHnY3cKoro MaTeMaTHKa).
B03hMeM KaKoH-HH6Y,l(h H3hIK (HanpHMep, PYCCKHH) 2),
cpe,l(CTBaMH KOToporo MO>KHO onHChIBaTh H Onpe,l(eJlHTh
Bce tlHCTO apHcpMemqeCKHe CBOHCTBa tlHCe.TI. PaCCMOT-
pHM onpe,l(e.TIeHIDI, KOTophIe MO)KHO CcpOPMYJIHpOBaTh
Ha 3TOM H3hIKe. JlCHO, tlTO HeKOTophIe TepMHHhI, OTHO-
CHInHeCH K apHcpMeTHtleCKHM CBOHCTBaM, HaM onpe,l(e-
JIHTh HBHhIM 06pa30M Bce paBHO He Y,l(aCTCH (c tlerO-TO
HMO HatlaTh H B onpe,l(e.TIeHIDIX BO H36e)KaHHe cHTYa-
u
nHH, H3BeCTHhIX no,l( Ha3BaHIDIMH «nopOtlHOrO Kpyra»
H «6eCKOHetlHOrO cnYCKa»), XOTH, KOHetlHO, MhI MO)KeM
B npHHnHne nOHHMaTh CMhICJI 3TH X CJIOB H 6e3 onpe,l(e-
JIeHHH. ,L{mI HameH ne.TIH Hecym:ecTBeHHo, KaKHe HMeHHO
TepMHHhI npHIDITh B KatleCTBe HCXO,l(HhIX, HeOnpe,l(eJlHe-
MhIX; MhI MO)KeM, HanpHMep, CtlHTaTh, tlTO MhI nOHHMaeM
u
CMhICJI npe,lVIO)KeHHH «neJIoe tlHCJIO ,l(e.TIHTCH Ha pyroe
ne.TIoe tlHCJIO», «ne.TIoe tlHCJIO HBJlHeTCH npoH3Be eHHeM
,l(BYX neJIhIX tlHCe.TI» H T. n. CBOHCTBO 6hITh npocThIM tlHC-
JIOM TOr,l(a MO)KHO Onpe,l(eJIHTh CJIe,l()'lOInHM 06pa30M:
«He ,l(eJIHThCH HH Ha O,l(HO ne.TIoe tlHCJIO, KpOMe caMO-
ro ce6H H tlHCJIa 1»; CBOHCTBO 6hITh TOtlHhIM KBMpa-
TOM: «6hITh npOH3Be,l(eHHeM HeKOToporo ne.TIoro tlHCJIa
Ha TO )Ke tlHCJIO» H T. n.
2) KOHeqHO, y aBTopOB peqb mna 06 aHrJIHHCKOM, a y caMoro PH-
mapa - 0 <ppaHIfj'3CKOM H3bIKe. - IIpUM.. nepea.
69
6. V1Ae51 KOAHpOBaHH5I H ee HcnOAb30BaHHe B MaTeMaTHKe
JIerKo B eTb, qTO K oe TaKoe onpe eJIeHHe COCTOHT
JIHllIb H3 KOHeqHOrO qHCJIa CJIOB, a nOToM)' H H3 KOHeq-
Horo qHCJIa 6)'KB an<l>aBHTa. ITOSTOMY MbI MO:>KeM BBeCTH
u
,lVUI TaKHX CJIOBeCHbIX onpe eJIeHHH OTHOllIeHHe nop -
Ka, CqHTaH O HO onpe eJIeHHe npe llIecTByID HM pyro-
eCJIH qHCJIO 6)'KB, H3 KOTOpbIX COCTOHT nepBoe onpe-
eJIeHHe, MeHbllIe qHCJIa 6)'KB, COCTaB.JI.HIO HX BTopoe
onpe eJIeHHe; B Tex :>Ke CJIyqaHX, Kor a Ba onpe eJIeHIDI
COCTO.HT H3 O Horo H Toro :>Ke qHCJIa 6)'KB 3), O HO H3 HHX
CqHTaTb npe llIeCTBYIO HM pyrOMY B 06bIqHOM JIeK-
CHKorpa<l>HqeCKOM (aJI<l>aBHTHoM, CJIOBapHOM) nop Ke.
I1CXOAH H3 TaKoro ynop OqeHIDI MO:>KHO Tenepb pacno-
JIO:>KHTb Bce onpe eJIeHIDI paCCMaTpHBaeMoro B a B no-
CJIe OBaTeJIbHOCTb, conOCTaBHB K OMY H3 HHX e HH-
u
CTBeHHoe HarypaJIbHOe qHCJIO - HOMep B STOH :>Ke nOCJIe-
OBaTeJIbHOCTH. Tor a caMoe KOpOTKOe (H CTO.H ee paHee
pyrHX B aJIcpaBHTHOM nop Ke) onpe eJI He nOJIyqHT
HOMep 1, CJIeAYIO ee 3a HHM B STOM «CJIoBflpe onpe e-
JIeHHH» - HOMep 2 H T. . /
ITOCKOJIbK)' Ka oMY onpe eJIeHHIO Tenepb COnOCTaBJIe-
HO HeKoTopoe HarypaJIbHOe qHCJIO, TO MO:>KeT OKa3aTbC.H,
qTO B HeKOTopbIX CJIyqaHX qHCJIO, COnOCTaBJIeHHOe KaKO-
M)'-HH6Y b onpe eJIeHHIO, caMO 6y eT 06JIaJ(aTb onpe e-
u
.JI.HeMbIM CBOHCTBOM.
* CH-ryaUH5I 3AeCb B T04HOCTH TaKOBa )Ke, KaK B TOM CAY4ae, KorAa Bce
CAoBa B 06bl4HOM opcporpacpH4eCKoM CAoBape AeA5ITC5I Ha ABa KAacca:
OAHOCAO)KHble H MHOrOCAO)KHble; npH 3TOM CAOBO «MHorOCAO)KHoe» caMo
oKa3blBaeTC5I MHOrOCAO)KHbIM.
3) n pOnyCK Me CJIOBaMH MO)KHO npH aTOM CQHTan OC060H «6yx-
BOH» (HanpHMep, nOCJIe,n;HeH B a.n:4>aBHTe) H.nH npoCTO nHcan CJIOBa
no,n;pH,lJ;, 6e3 nponycKOB. - npUM. nepea.
70
6. V1 e51 KOAHpOBaHH5I H ee HcnOAb30BaHHe B MaTeMaTHKe
ITYCTb, HanpHMep, onpe eJUllO ee BblproKeHHe «He e-
JIHTbC.H HH Ha O HO HaTypaJIbHOe qHCJIO, KpOMe caMO-
ro ce6.H H qHCJIa 1» OKa3aJIOCb B HallieH nOCJIe OBa-
TeJIbHOCTH Ha 1 7 -M MeCTe; .HCHO, qTO COnOCTaBJIeHHOe
eMY qHCJIO 1 7 caMO no naJ(aeT no 3TO onpe eJIeHHe.
ITYCTb, C pyrOH CTOpOHbI, onpe eJUllO ee BblpIDKeHHe
«6bITb npoH3Be eHHeM HeKoToporo HaTypaJIbHOrO qHC-
JIa Ha TO )Ke caMoe qHCJIO» noJIyqH.7IO HOMep 15; caMO
"llHCJIO 15, OqeB HO, He .HB.JI..HeTC.H TOqHbIM KBaJ(paTOM
H nOTOM)' aHHbIM CBOHCTBOM He 06JIaJ(aeT. Ha30BeM
qHCJIa, He 06JIaJ(aIOInHe CBOHCTBaMH, onpe eJUleMbIMH
npe,n.nO)KeHIDIMH, KOTOpbIM OHH COOTBeTCTBYIOT B onH-
caHHOH HaMH HYMepa HH, pUWap06b1MU. TaKHM o6pa-
30M, «x - pHllIapOBO qHCJIO» - 3TQ npOCTO COKpa e-
HHe BblpIDKeHIDI «x He 06JIaJ(aeT CBOHCTBOM, onpe e-
.JI..HeMbIM npe,n.nO)KeHHeM, HMelO HM HOMep x B aHHoH
u
CJIOBapHOH nOCJIe OBaTeJIbHOCTH Onpe eJUllO HX npe -
JIO)KeHHH». (CKIDKeM, qHCJIO 17 H3 Halliero nepBoro npH-
Mepa He .HB.1I.HeTC.H pHllIapOBbIM, a qHCJIO 15 H3 BToporo
npHMepa - pHllIapoBo.)
Tenepb MbI }')Ke MO)KeM CcpOpM)'JIHpOBaTb napaJ(oKc PH-
llIapa. Onpe eJUllO ee BblpIDKeHHe .wrn CBoHcTBa 6bITb
pHllIapoBbIM qHCJIOM onHCbIBaeT, OqeBH HO, HeKoTopoe
apHcpMeTHqeCKOe CBOHCTBO HaTypaJIbHbIX qHCeJI. 3Ha-
qHT, caMO onpe eJUllO ee BblpIDKeHHe BXO HT B onH-
caHH)'IO BbIllie nOCJIe OBaTeJIbHOCTb Onpe emIlO HX BbI-
proKeHHH. Ho TOr a OHO HMeeT B 3TOH nOCJIe OBaTeJIb-
HOCTH HeKoTopbIH HOMep, KOTOpbIH MbI 0603HaqHM qe-
pe3 n. 3aJ(aJ(HM Tenepb BnOJIHe eCTecTBeHHbIH Bonpoc
(HeMe,n.neHHo npHBOA.H HH K aHTHHOMHH PHllIapa): .HB-
JUleTC.H JIH qHCJIO n pHllIapoBbIM? qHTaTeJIb, KOHeqHO,
71
6. V1Ae51 KOAHpOBaHH5I H ee HcnOAb30BaHHe B MaTeMaTHKe
cpa3Y yBH,nHT, qTO npoTHBOpeqHe Tenepb HeH36e:>KHo.
B caMOM eJIe, qHCJIO n .HBJUleTC.H pHmapoBbIM B TOM
H TOJIbKO B TOM CJIyqae, eCJIH OHO He 06JIaJ(aeT CBOHCTBOM,
onHCbIBaeMbIM npe,n.nO)KeHHeM, HMelO HM HOMep n, T. e.
He 06JIaJ(aeT CBOHCTBOM 6bITb pHmapOBbIM! Kopoqe ro-
BOp.H, n pHmapOBO Tor a H TOJIbKO Tor a, Kor a OHO
He pHmapOBO, T. e. yrBep>K.ZJ:eHHe «n - pHmapOBO qHC-
JIO» .HBJUleTC.H O HOBpeMeHHO HCTHHHbIM H JIO)KHbIM.
CJIe eT 3aMeTHTb, qTO 3TO npOTHBOpeqHe B H3BeCT-
HOM CMbICJIe eCTb TpIOK, KOTOpbIH HaM Y aJIC.H 6JIaro-
ap.H He BnOJIHe TOqHOMY C06JIIO eHHIO npaBHJI HrpbI.
,[(eJIO B TOM, qTO MbI cpaKTHqeCKH HCnOJIb30BaJIH O HO
ony eHHe, KOTopoe, O HaKO, npe nOqJIH B .HBHOM BH-
e He cpOPMYJIHpOBaTb. MbI COrJIaCHJIHCb paCCMaTpHBaTb
onpe eJIeHIDI 1.Lucmo apUfPMemU1.LeCKUX CBOH B HaTY-
paJIbHbIX qHCeJI, T. e. CBOHCTB, CPOPMYJIHpyeMbI B Tep-
MHHax TaKHX nOIDITHH, KaK apHcpMeTHqeCKOe O:>KeH e,
YMHO:>KeHHe H T. n. 3aTeM, O HaKO, 6e3 OnOJIHHTeJIb-
HbIX orOBopOK MbI BKJIlOqHJIH B TY :>Ke nOCJIe OBaTeJIb-
HOCTb onpe eJIeHHH npe,n.nO)KeHHe, CCPOPMYJIHpOBaHHoe
nocpe CTBOM ynoMHHaHIDI 0 HeKOTopOM cnoco6e aanucu
apHcpMeTHqeCKHX CBOHCTB. CTporo rOBopH, onpe eJIeHHe
CBOHCTBa 6bITb pHmapOBbIM qHCJIOM npOCTO He npHHaJ(-
u u
JIe:>KHT K TOH nOCJIe OBaTeJIbHOCTH onpe eJIeHHH, KOTO-
pM BHaqaJIe onHCbIBaJIaCb, TaK KaK 3TO onpe eJIeHHe
HCnOJIb3yeT TaKHe MeTaMaTeMaTHqeCKHe nOIDITIDI, KaK
HOMep 6)'KBbI (HJIH Bo06m:e 3HaKa) B HeKOTopOH nOCJIe-
OBaTeJIbHOCTH. TaKHM 06pa30M, eCJIH MbI 6y eM qeTKO
pa3JIHqaTb yrBep>K.ZJ:eHIDI CQMOU apHcpMeTHKH (OTHOC.H-
HeC.H K qHCJIaM, a OTHIO b He K 3anHC.HM, B KOTopbIe
TaKHe qHCJIa BXOAHT, T. e. K paBeHCTBaM, HepaBeHCTBaM
H Bo06 e cpopMYJIaM) H yrBep>K.ZJ:eHIDI omHocume.n.bHO
72
6. V1Ae51 KOAHpoBaHHSI H ee HcnOAb3(jBasHHe B MaTeMaTHKe
apH<pMeTHKH (T. e. KaK pa3 yrBep eHIDI 06 apH<pMeTH-
'IeCKHX CPOPMYJIax), TO Mbl He nOJIy'IHM HHKaKoro napa-
OKca PHmapa.
3Ha'IHT, caM no ce6e nap3J(OKC PHmapa COBceM He cTpa-
meH. Ho caMa cxeMa npHBOAH ero K HeMY pacc e-
HHH 'Ipe3BW'IaHHO noyqHTeJIbHa H nJIO OTBopHa. Pe'Ib
H,l(eT 0 B03MO)KHOCTH «oT06pIDKeHHH» (HJIH «nepeBO-
_ a») MeTaMaTeMarn'IeCKHX BbICKa3bIBaHHH, OTHOC.H HX-
CH K HeKOTopOH OCTaTO'IHO 60raTOH <pOpMaJIbHOH CH-
CTeMe, B caMY CHCTeMY. CaMa no ce6e H e.H «nepeBO a»
u
xopomo H3BeCTHa H HrpaeT BIDKHeHIII)'IO pOJIb BO MHO-
rHX 06JIaCT.HX MaTeMaTHKH. TaKaH H e.H JIe)KHT, HanpH-
u u
Mep, B OCHOBe BceH aHaJIHTH'IeCKOH reoMeTpHH, r e reo-
MeTpH'IeCKHe nOHHTHH nepeBoAHTC.H B aJIre6paH'IeCKHe
(apH<pMeTH'IeCKHe), TaK 'ITO BMeCTO reoMeTpH'IeCKHX co-
OTHOmeHHH Mbl, no cym;eCTBY, HMeeM eJIO C aJIre6paH-
'IeCKHMH. (BcnoMHHTe XOT.H 6bl 06c aBllIeeC.H B pa3 e-
JIe 2 oT06pIDKeHHe reoMeTpHH B aJIre6py; HCnOJIb30BaH-
Hoe fHJIb6epTOM AR.H OKa3aTeJIbCTBa OTHOCHTeJIbHOH
HenpOTHBOpe'IHBOCTH ero CHCTeMbl reOMeTpH'IeCKHX aK-
CHOM). TaKHe oT06pIDKeHHH-«nepeBO bl» HrpalOT 60JIb-
III)'IO pOJIb H B <pH3HKe, HanpHMep, Kor a CBOHCTBa 3JIeK-
TpH'IeCKOrO TOKa H3JIaraIOTC.H Ha .H3blKe rH.npO HHa H-
'IeCKHX .HBJIeHHH. 9Ta )Ke H e.H nepeBo a JIe)KHT B OCHOBe
TeXHH'IeCKOrO MO eJIHpOBaHHH - H eT JIH pe'Ib 06 HC-
CJIe OBaHHH CBOHCTB MOAeJIH CaMOJIeTa (HJIH )Ke caMO-
JIeTa B HaTYPaJIhHYIO BeJIH'Iliror) B a3po HHaMH'IeCKOH
Tpy6e, HJIH )Ke 06 H3yqeHHH B JIa60paTopHblX YCJIOBIDIX
pacnpe eJIeHHH KaKHX-JIH60 MaTepHaJIbHblX Macc C no-
u
MOII(blO aHaJIOrOBOH MO eJIH, r e pOJIh 3THX Macc HrpalOT
3JIeKTpH'IeCKHe 3ap.H.nbl.
73
6. V1Ae51 KOAHpOBaHH5I H ee HcnOAb30BaHHe B MaTeMaTHKe
JIerKo BH eTb, qTO K oe TaKoe onpe eJIeHHe COCTOHT
JIHllIb H3 KOHeqHOrO qHCJIa CJIOB, a nOToMY H H3 KOHeq-
Horo qHCJIa 6YKB ancpaBHTa. ITOSTOMY MbI MO:>KeM BBeCTH
u
AJUI TaKHX CJIOBeCHbIX onpe eJIeHHH OTHOllIeHHe nOPM-
Ka, CqHTaH O HO onpeAeJIeHHe npe llIecTByID HM pyro-
MY, eCJIH qHCJIO 6YKB, H3 KOTOpbIX COCTOHT nepBoe onpe-
eJIeHHe, MeHbllIe qHCJIa 6yKB, COCTaBJIHIO HX BTopoe
onpe eJIeHHe; B Tex :>Ke CJIyqaHX, Kor a Ba onpe eJIeHIDI
COCTO.HT H3 O Horo H Toro :>Ke qHCJIa 6YKB 3) , O HO H3 HHX
CqHTaTb npe llIeCTByID HM pyrOMY B 06blqHOM JIeK-
CHKorpacpHqeCKOM (ancpaBHTHOM, CJIOBapHOM) nop Ke.
I1CXOAH H3 TaKoro ynoPMOqeHIDI MO)KHO Tenepb pacno-
JIO:>KHTb Bce onpe eJIeHIDI paCCMaTpHBaeMoro B a B no-
CJIe OBaTeJIbHOCTb, conOCTaBHB K OMY H3 HHX e HH-
u
CTBeHHoe HarypanbHoe qHCJIO - HOMep B STOH )Ke nOCJIe-
OBaTeJIbHOCTH. Tor a caMoe KopOTKoe (H CTO.H ee paHee
pyrHX B aJI<l>aBHTHoM nop Ke) onpe eJIeHHe nOJIyqHT
HOMep 1, CJIeAYIO ee 3a HHM B STOM «CJIOBape onpe e-
JIeHHH» - HOMep 2 H T. .
TIOCKOJIbK)' Ka oMY onpe eJIeHHIO Tenepb COnOCTaBJIe-
HO HeKoTopoe HarypaJIbHOe qHCJIO, TO MO)KeT OKa3aTbC.H,
qTO B HeKOTopbIX CJIyqaHX qHCJIO, COnOCTaBJIeHHOe KaKO-
MY-HH6Y b onpe eJIeHHIO, caMO 6YAeT 06JIaJ(aTb onpe e-
u
JIHeMbIM CBOHCTBOM.
* CHT}'aUH5I 3AeCb B T04HOCTH TaKoBa )Ke, KaK B TOM CA}'4ae, KorAa Bce
CAoBa B 06bl4HOM opcporpacpH4eCKoM CAoBape AeA5ITC5I Ha ABa KAacca:
OAHOCAO)KHble H MHOrOCAO)KHble; npH 3TOM CAOBO «MHorOCAO)KHoe» caMo
oKa3blBaeTC5I MHOrOCAO)KHbIM.
3) II pOnyCK Me CJIOBaMH MO>KHO npH aTOM CQHTan OC060H «6YK-
BOR» (HanpHMep, nOCJIe.n;HeR B a.n4>aBHTe) HJIH npOCTO nHcaTb CJIOBa
no.n;pH,lJ;, 6e3 nponycKOB. - npUM. nepea.
70
6. AeSi KOAHpoBaHHSI H ee HcnOAb30BaHHe B MaTeMaTHKe
ITYCTb, HanpHMep, onpeAeJUllO ee BbIpIDKeHHe «He Ae-
JIHTbC.H HH Ha OAHO HaTYPaJIbHOe qHCJIO, KpOMe caMO-
ro ce6.H H qHCJIa 1» OKa3aJIOCb B HameH nOCJIeAOBa-
TeJIbHOCTH Ha 1 7 -M MeCTe; .HCHO, qTO COnOCTaBJIeHHOe
eMY qHCJIO 1 7 caMO nOAnaJ(aeT nOA 3TO onpeAeJIeHHe.
ITYCTb, C APyrOH CTOpOHbI, onpeAeJUllOInee BbIpa:>KeHHe
«6bITb npoH3BeAeHHeM HeKoToporo HaTYPaJIbHOrO qHC-
JIa Ha TO )Ke caMoe qHCJIO» nOJIyqHJIO HOMep 15; caMO
qHCJIO 15, OqeBHAHo, He .HB.IDIeTC.H TOqHbIM KBaJ(paTOM
H nOTOM)' AaHHbIM CBOHCTBOM He 06JIaJ(aeT. Ha30BeM
QHCJIa, He 06JIaJ(aIOInHe CBOHCTBaMH, onpeAeJUleMbIMH
npe,n.nO)KeHIDIMH, KOTOpbIM OHH COOTBeTCTB)'IOT B onH-
caHHOH HaMH HYMepa HH, pUWapOBbIMU. TaKHM o6pa-
30M, «x - pHmapOBO qHCJIO» - 3TQ npOCTO COKpa e-
HHe BbIpIDKeHIDI « X He 06JIaJ(aeT CBOHCTBOM, onpeAe-
.IDIeMbIM npe,n.nO:>KeHHeM, HMelOInHM HOMep x B AaHHOH
u
CJIOBapHOH nOCJIeAOBaTeJIbHOCTH OnpeAeJUllO HX npeA-
JIO:>KeHHH». (CKIDKeM, qHCJIO 17 H3 Hamero nepBoro npH-
Mepa He .HB.IDIeTC.H pHmapOBbIM, a qHCJIO 15 H3 BTOporo
npHMepa - pHmapoBo.)
Tenepb MbI }')Ke MO:>KeM C<pOpM)'JIHpOBaTb napaJ(oKc PH-
mapa. OnpeAeJUllOInee BbIpIDKeHHe ,n.n.H CBoHcTBa 6bITb
pHmapoBbIM qHCJIOM on HCbIBaeT, OqeBHAHO, HeKoTopoe
apH<pMeTHqeCKOe CBOHCTBO HaTYPaJIbHbIX qHCeJI. 3Ha-
qHT, caMO OnpeAemIlO ee BbIpIDKeHHe BXOAHT B onH-
caHHYIO BbIme nOCJIeAOBaTeJIbHOCTb OnpeAeJUllO HX BbI-
pIDKeHHH. Ho TOrAa OHO HMeeT B 3TOH nOCJIeAOBaTeJIb-
HOCTH HeKoTopbIH HOMep, KOTOpbIH MbI 0603HaqHM qe-
pe3 n. 3aJ(aJ(HM Tenepb BnOJIHe eCTeCTBeHHbIH Bonpoc
(HeMe,n.neHHo npHBO,lJ)l HH K aHTHHOMHH PHmapa): .HB-
.IDIeTC.H JIH qHCJIO n pHmapoBbIM? qHTaTeJIb, KOHeqHO,
71
6. VlAe51 KOAHpOBaHI451 H ee HcnOhb30BaHHe B MaTeMaTHKe
Ha H ee oTo6pIDKeHIDI 4) OCHOBaH TaK Ha3bIBaeMbIH npHH-
u u
:QHn BOHCTBeHHOCTH B npOeKTHBHOH reOMeTpHH, COCTO-
u u
j[[ HH B B03MO)KHOCTH B3aHMHOH 3aMeHbI B aKcHoMax
(a 3HaQHT, H B TeopeMax) npOeKTHBHOH reOMeTpHH Tep-
MHHOB «TOqKa» H «npHMaH», B pe3YJIbTaTe qero aKCHO-
MbI nepeXO.lVlT B aKCHOMbI (COOTBeTCTBeHHO TeopeMbI -
B TeopeMbI). ITpH O HOM H3 TaKHX «nepeBO OB» CJIO-
BO «TOqKa» MO)KHO CqHTaTb «06pa30M», CJIOBO «npH-
MaH» - «npo06pa30M»; npH 06paTHOM nepeBo e pOJIH
MeIDIIOTCH.
CaMbIM cymecTBeHHbIM B 06c aeMOH 3 eCb H ee «MO-
eJIHpOBaHIDI» HBJUleTCH TO, qTO a6CTpaKTHaH cTpYKTYPa
u u
OTHOIIleHHH, BbInOJIIDIeMbIX ,lVI.H «npe,nMeTOB» KaKOH-
JIH60 06JIaCTH, MO)KeT 6bITb H3yqeHa C nOMom:blO pac-
u
CMOTpeHIDI OTHOIIleHHH, HMelOID:HX MeCTO Me «npe -
MeTaMH» (KaK rrpaBHJIO, pyrOH npHpO bI, qeM «npe -
MeTbI» HCXO HOH 06JIaCTH), npHHa,wIe)Kam:HMH COBceM
pyrOH 06JIaCTH. I1MeHHo 3Ta H eH «KO HpOBaHIDI» JIe-
)KHT B OCHOBe ,l(OKa3aTeJIbCTBa re eJI.H. ECJIH HeKOTopbIe
- CJIO)KHbIe MeTaMaTeMaTHqeCKHe BbICKa3bIBaHIDI 0 <l>op-
MaJIH30BaHHOH CHCTeMe apH<}:>MeTHKH MO)KHO, KaK pac-
CqHTbIBaeT re eJIb, nepeBeCTH (RJIH «oT06pa3HTb») B He-
KOTopbIe apH<pMeTHqeCKHe BbICKa3bIBaHIDI, npHHa,wIe-
)Ka He caMOH CHCTeMe, TO 3TO }')Ke caMO no ce6e HBHT-
C.H 60JIbIIlHM ,l(OCTH)KeHHeM B eJIe pa3BHTIDI Teopem-
u
KO- OKa3aTeJIbCTBeHHOH TeXHHKH, TaK )Ke, KaK HCCJIe O-
4) MO:>KHO 6bIJIO 6bI CKa3an «nepeBO,ZJ;a», «MO,ZJ;eJIHpOBaHHH», «KO-
,ZJ;HpOBaHHH», «npe,ZJ;CTaBJIeHHH»; B nepeBO,ZJ;e MbI ,ZJ;a.nee 6Y,ZJ;eM C03Ha-
TeJIbHO BapbHpOBaTb ynOTpe6JIeHHe aTHX TepMHHOB, 'IT06bI nO,ZJ;qepK-
HYTb npHHI(HnHaJIbHOe pO,ZJ;CTBO nOHHTHH, BbIpIDKaeMbIX aTHMH Tep-
MHHaMH, Me C060H H C HCnOJIb3yeMbIM ,ZJ;aJIee nOHHTHeM «HyMepa-
nHH». - npUM. nepeB.
74
6. V1Ae51 KOAa.1pOBaHa.151 a.1 ee a.1CnOAb30BaHa.1e B MaTeMaTa.1Ke
BaTb aJIre6paHqeCKHe cooTHomeHIDI, npeACTaBJUlIO He
(H306pIDKalO He, KOAHpyIO He) HeKoTopbIe reoMeTpH-
qeCKHe COOTHomeHIDI Me KpHBbIMH H nOBepXHOCT.H-
MH, ropa3Ao YA06Hee, qeM HMeTb AeJIO C caMHMH reo-
MeTpHqeCKHMH cOOTHOmeHIDIMH, - TOqHO TaK )Ke apHcp-
MeTHQeCKHe aHaJIOrn (<<06pa3bI») CJIO)KHbIX JIOrHQeCKHX
cooTHomeHHH OKa3bIBaIOTC.H B H3BeCTHOM CMbICJIe 60JIee
0603pHMbIMH H AOCTyIIHbIMH ,lVI.H H3yqeHIDI, qeM HX JIO-
rHqeCKHe «npo06pa3bI».
IIcnoJIb30BaHHe HAeH KOAHpoBaHIDI, KaK MbI )"Ke OT-
MeqaJIH, JIe)KHT B OCHOBe 3HaMeHHToH pa60TbI reAeJI.H.
CJIeA)'.H cxeMe pacc)"K,neHIDI, OqeHb 6JIH3KOH K TOH, QTO
npoBoAHTCH B napMoKce PHmapa (HO ycoBepmeHcTBY.H
ee npH 3TOM TaKHM 06pa30M, qTO OHa CTaHOBHTC.H He-
Y.H3BHMOH no OTHomeHHIO K C<POPMYJIHpOBaHHbIM BbIme
KpHTHqeCKHM 3aKJIlOqeHIDIM), reAeJIb nOKa3bIBaeT, qTO
MeTaMaTeMaTHqeCKHe BbICKa3bIBaHIDI 06 apH<pMeTHqe-
CKOM cpOpMaJIH30BaHHoM HCQHCJIeHHH MO)KHO npe CTa-
BHTb nocpeACTBOM HeKOTopbIX apHcpMeTHqeCKHX cpop-
MYJI BHympu HCqHCJIeHIDI. KaK MbI nOKIDKeM nOAP06Hee
u u
B CJIe,l\YlO eM pa3AeJIe, eM)' YAaJIOCb HaHTH TaKOH Me-
TOA apH<pMeTHqeCKOrO KOAHpOBaHIDI MeTaMaTeMaTHqe-
CKHX BbIcKa3bIBaHHH, qTO,lVI.H HeKoTopoH CPOPMYJIbI, BbI-
u
pIDKalO eH HCTHHHoe MeTaMaTeMaTHqeCKOe yrBep e-
HHe 0 <POPMYJIax apHcpMeTHKH, HH OHa caMa, HH ee OT-
pH aHHe He AOKa3yeMbI B cpOpMaJIbHOH apHcpMeTHKe.
ITOCKOJIbK)' OAHa H3 3THX cpOpM)'JI, BbIpIDKalO a.H HC-
THHHoe apH<}:>MeTHQeCKOe BbIcKa3bIBaHHe, He BbIBO H-
Ma H3 apHcpMeTHqeCKHX aKCHOM, TO aKCHOMbI 06pa3Y-
lOT HenOJIHyIO CHCTeM)'. ITpe)VIO)KeHHbIH reAeJIeM MeTO
KOAHpOBaHIDI n03BOJIHJI eM)' TaIOKe nOCTpOHTb apHcp-
MeTHqecKyIO <pOpM)'JIY, COOTBeTCTBYIOInYIO MeTaMaTeMa-
75
6. V1AeSi KOAHpoBaHHSI H ee HcnOAb30BaHHe B MaTeMaTHKe
THqeCKOMY BbICKa3bIBaHHIO «apH<pMeTHqeCKOe HCqHCJIe-
HHe HenpoTHBOpeqHBo», H nOKa3aTb, qTO STa <popMYJIa
He oKa3yeMa B (STOM )Ke!) apHcpMeTHqeCKOM HCqHCJIe-
HHH. OTclO a CJIe eT, qTO ynoM.HHYToe MeTaMaTeMaTH-
qeCKOe BbICKa3bIBaHHe He MO)KeT 6bITb YCTaHOBJIeHO 6e3
npHBJIeqeHIDI HeKOTopbIX AOnOJIHHTeJIbHbIX eAYKTHB-
HbIX cpe CTB, He npe CTaBHMbIX (T. e. He KO HpyeMbIx,
He nepeBO HMbIX) B caMOM apH<pMemqeCKOM HCqHCJIe-
HHH, TaK qTO eCJIH STO BbICKa3bIBaHHe H MO)KHO AOKa3aTb,
TO Y'K 3aBeAOMO C npHBJIeqeHHeM cpeACTB, HenpOTHBope-
qHBOCTb KOTOpbIX He MeHee COMHHTeJIbHa, He)KeJIH caMa
no ce6e HenpOTHBOpeqHBOCTb apH<pMeTHKH. Bce BIDI(-
HeHIlIHe BbIBO,n:bI 6bmH no.nyqeHbI reAeJIeM C HCnOJIb-
u u u
30BaHHeM npH aHHOH HM qpe3BbIqaHHO OcrpO)'MHOH
CHCTeMbI qHCJIOBOrO KO HpOBaHIDI, HJIH, KaK MbI 6y eM
,n:a.nee rOBopHTb, HY.Mepaquu.
7
TeopeMbl reAeMi
7. 1. r eAeAeBCKa51 HYMepauH5I
reAeJIb npe e Bcero onHCa.n HeKOTopoe cpOpMaJIH30-
BaHHoe HC'IHCJIeHHe, cpeACTBaMH KOToporo MO)KHO BbI-
pa3HTb Bce 06bI'IHble apH<pMeTH'IeCKHe nOIDITIDI H ycra-
HOBHTb H3BeCTHble apHcpMeTH'IeCKHe COOTHOllieHIDI.
reAeAb HcnOAb30BaA HeCKOAbKO ynpoweHHblt1 BapHaHT CHCTeMbl, onHcaH- +
HOt1 B Principia Mathematica. Ho AASI ero ueAH T04HO TaK )Ke nOAXOAHT
AI060e HC4HCAeHHe, B KOTOpOM MO)KHO nocTpoHTb cHcTeMY HaTypaAbHblX
4HceA C onpeAeAeHHblMH Ha Het1 apHcpMeTH4ecKHMH onepauHSlMH.
OPMYJIbI 3Toro HC'IHCJIeHIDI CTpOHTCH HCXOAH H3 HeKO-
Toporo 3anaca 3JIeMeHTapHbIx CHMBOJIOB, 06pa3yID Hx
a.n<paBHT cHcTeMbI. B 3TOM HC'IHCJIeHHH, KaK 06bI'IHO, BbI-
AeJIeHO HeKoTopoe MHO)KeCTBO HCXOAHbIX <POPMYJI (aK-
CHOM) H TO'IHO nepeQHCJIeHbI npaBH.7Ia npeo6pa30BaHIDI
(npaBHJIa BbIBOAa), nocpeAcrBoM KOTOpbIX H3 aKCHOM
BbIBOAHTCH TeOpeMbI.
reAeJIb nOKa3aJI, 'ITO Ka OMY 3JIeMeHTapHOMY CHM-
BOJIY, KIDKAOH <popMYJIe (T. e. n:enO'IKe 3JIeMeHTap-
77
7. TeopeMbl reAeASI
HbIX CHMBOJIOB) H K OMY OKa3aTeJIbCTBY (KOHe1.IHOH
nOCJIe OBaTeJIbHOCTH <POPMYJI) MO>KHO OOH03HU1.LHblM
o6pa30M npHnHcaTb HeKOTOpbIH HOMep (HarypaJIbHOe
1.IHCJIO). TaKoH HOMep, CJIY>Kam:HH CBoero po a 3Ha1.I-
u
KOM, HpJIbIKOM, YKa3bIBalOm:HM Ha OTMe1.IaeMbIH HM
06'beKT - CHMBOJI, <POPMYJIY HJIH oKa3aTeJIbCTBO -
<pOpMaJIbHOH cHcTeMbI, MbI 6y eM Ha3bIBaTb «re eJIeB-
CKHM HOMepoM» 3Toro CHMBOJIa, <POPMYJIbI HJIH oKa-
3aTeJIbCTBa 1) .
9JIeMeHTapHbIe CHMBOJIbI, COCTaBJIHIOm:He aJI<paBHT CH-
CTeMbI, 6bIBalOT BYX COpTOB: KOHCTaHTbI H fepeMeH-
HbIe. MbI 6y eM C1.IHTaTb, 1.ITO Y HaC eCTb pOBHO -.zteCHTb
CHMBOJIOB-KOHCTaHT, KOTOpbIM MbI npHnHmeM B Ka1.Ie-
CTBe re eJIeBCKHX HOMepOB 1.IHCJIa OT 1 o 10. TI01.ITH Bce
3TH CHMBOJIbI 1.IHTaTeJIlO )')Ke H3BeCTHbI: «IV» (COKpam:e-
HHe ,lVUI «He»), «V» (<<HJIH»), «::>>> (<<eCJIH..., TO...»),
«=» (<<paBHO»), «0» ( H<PPOBOH 3HaK, H306pIDKalOm:HH
1.IHCJIO «H)'JIb»), a TaIOKe TpH «3HaKa npenHHaHH.H»: JIe-
BaH cKo6Ka «(», npaBaH cKo6Ka «)>> H 3aIDITaH «,». Kpo-
Me Toro, HaM nOH o6HTCH em:e Ba CHMBOJIa: nepeBep-
HYTaH 6YKBa «3 » (1.IHTaeMaH KaK «cym:eCTByeT» H Ha-
3bIBaeMaH «KBaHTopoM cym:ecTBoBaHIDI») H CTp01.IHaH
JIaTHHCKaH 6YKBa «8», 0603Ha1.IalOm:aH 1.IHCJIOBOH one-
u
paTOp, conOCTaBJUlIOm:HH K OMY HarypaJIbHOMY 1.IHC-
JIY Henocpe CTBeHHO CJIe m:ee 3a HHM 1.IHCJIO. TIpH-
Mep: <POPMYJIY «3 x (x = sO)>> MO>KHO np01.IeCTb KaK
«cym:ecTBYeT TaKoe x, 1.ITO x Henocpe CTBeHHo CJIe -
eT 3a 1.IHCJIOM 0». BbInHmeM Bce HCnOJIb3yeMbIe HaMH
1) HMeeTCH MHoro pa3JIHlIHbIX cnoco60B npHnHCbIBaHHH re e-
JIeBCKHX HOMepOB, H KaKOH H3 HHX BbI6paTb - COBepmeHHO Hecy-
m;eCTBeHHO.
78
7. 1. r eAeAeBcKaSl HYMepauHSI
CHMBOJIbI-KOHCTaHThI (no HHMH YKa3aHbI COOTBeTCTBY-
IOm:He re eJIeBCKHe HOMepa):
rv V ::> 3 - 0 s ( )
1 2 3 4 5 6 7 8 9 10
KpOMe 3JIeMeHTapHbIX CHMBOJIOB-KOHCTaHT, B aJI<paBHT
Halliero Hct.JHCJIeHHH BXO,lVlT em:e nepeMeHHbIe, npHt.IeM
nepeMeHHbIe Tpex COpTOB: 1.LUCIlOBbLe nepe.MeHHbLe «x»,
«y», «z» H T. . (BMeCTo HHX MO>KHO nO CTaBJUlTb «nH<p-
phI» H COCTaBJIeHHbIe H3 HHX (H t.IHCJIOBbIX nepeMeHHbIX)
«apH<pMemt.IeCKHe BhIpIDKeHHH», BbIpIDKalOm:He Hary-
paJIbHhIe t.IHCJIa); npono3uquOHaJlbHbLenepe.MeHHbLe «p»,
«q», «r» H T. . (BMeCTO HHX MO>KHO nO CTaBJI.HTb «<pOp-
MYJIbI», BhIpIDKalOm:He BbICKa3hIBaHHH); H, HaKOHen, npe-
OUKamHbLe nepe.MeHHbLe «P», «Q», «R» H T. . (BMeCTO
HHX MO>KHO nO CTaBJUlTb apH<pMeTHt.IeCKHe «npe HKa-
u
TbI», BbIpIDKalOm:He TaKHe CBOHCTBa H OTHOllieHHH, KaK
«60JIbllie t.JeM», «npOCTOe (t.IHCJIO)>> H T. n.). TIepeMeH-
..
HbIM TaIOKe conocraBJI.HIOTCH re eJIeBCKHe HOMepa, npH-
t.IeM eJIaeTCH 3TO B COOTBeTCTBHH CO CJIe m:HMH CO-
rJIallieHHHMH:
1) pa3.JIHt.IHhIM t.IHCJlOBhIM nepeMeHHbIM npHnHcbIBa-
IOTCH pa3JIHt.IHbIe npocThIe t.IHCJIa, 60JIblliHe 10;
2) pa3.7IHt.JHhIM npono3HnHOHaJIbHbIM nepeMeHHbIM npH-
nHCbIBaIOTCH KB paTbI pa3JIHt.IHbIX npoCTbIx t.IHCeJI,
60JIblliHX 10;
3) pa3.7IHt.JHhIM nperotKaTHbIM nepeMeHHbIM npHnHcbIBa-
IOTCH K)'6bI Pa3JIHqHbIX npOCThIX t.IHCe.n, 60JIblliHX 10.
79
7. TeopeMbl reAeASI
npHMep
BLIpa>KeHH1I,
re eJIeBCKHH KOTopoe
nepeMeHHaJI pa3pemaeTCH
HOMep
nO CTaBJllITh
BMeCTO aHHoH
u
nepeMeHHoH
x 11 0
Y 13 sO
z 17 y
. . .
. . .
. . .
P 11 2 0=0
q 13 2 3 X (x = sy)
Z 17 2 p-:Jq
. . .
. . .
. . .
P 11 3 «npOCTOe» *
Q 13 3 «COCTaBHoe» *
R 17 3 «60JILme qeM»*
* KaBLIqKH (,no6aBJIeHHLIe npH nepeBO e) 03HaqalOT 3 ecL,
qTO nO CTaBHTh MO>KHO He caMO HanHcaHHoe B npaBOM
CTOJI6n;e CJIOBO, a ern cpOpMaJILHyIO 3anHCL Ha 1I3LIKe Ha-
mero HCQHCJIeHH1I. - IIpUM.. nepeB.
B03bMeM KaKyro-HH6Y b <POPMYJIY HameR cHCTeMbI, Ha-
npHMep
3 x (x = SY)
80
7.1. reAeAeBcKaSl HYMepauHSI
(KOTOPyro MO)KHO np01.IeCTb KaK «c eCTByeT TaKoe x,
1.ITO x Henocpe CTBeHHo CJIeA)'eT 3a Y» H KOTOpaH BbIpa-
>KaeT TO 06CTOHTeJIbCTBO, 1.ITO ,lVUI K OrO 1.IHCJIa eCTb
HenOCpe CTBeHHO CJIe.nyIO ee 3a HHM t.JHCJIO). BblnHmeM
no K bIM H3 BXOAH HX B Hee CHMBOJIOB ero (CHMBO-
JIa) re eJIeBCKHH HOMep:
x
(
{-
8
{-
11
{-
5
{-
7
3
{-
4
x
s
Y
{-
13
)
{-
9
{-
11
KOHe1.IHO, HaM 6bI XOTeJIOCb conOCTaBHTb K OH <pop-
M)'JIe He Ha60p HOMepoB (KaK nOKa nOJIyqHJIOCb), a O HH-
e HHCTBeHHbIH onpe eJIeHHbIH HOMep. Ho 3TO Ot.JeHb
JIerKO C eJIaTb. A HMeHHO: conOCTaBHM 3TOH <popMYJIe
npoH3Be eHHe nepBblx BOCbMH npOCTbIX 1.IHCeJI B nopH -
Ke HX BeJIH1.IHHbI, npH1.IeM K oe H3 HHX B CTeneHH,
U ..
nOKa3aTeJIb KOTOpOH paBeH re eJIeBcKoM)' HOMepy COOT-
BeTCTByro ero 3JIeMeHTapHoro CHMBOJIa:
2 4 x 3 11 X 58 X 7 11 X 11 5 X 13 7 x 1 7 13 X 9 9 .
0603Ha1.IHM 3TO t.JHCJIO, CKa>KeM, 1.Iepe3 m. T01.IHO Ta-
KHM >Ke 06pa30M MbI MO>KeM npHnHcaTb B Ka1.IeCTBe
..
re eJIeBCKOro HOMepa e HHCTBeHHoe BnOJIHe onpe eJIeH-
U U
Hoe HarrypaJIbHOe t.JHCJIO K OH KOHe1.IHOH nOCJIe OBa-
TeJIbHOCTH 3JIeMeHTapHbIX CHMBOJIOB, B 1.IaCTHOCTH KIDK-
OH <popM)'JIe, npH1.IeM 1.IHCJIO npOCTbIX COMHO>KHTe.neH
TaKoro 1.IHCJIa Bcer a 6y eT paBHO 1.IHCJIY Bxo eHHH 3JIe-
MeHTapHbIX CHMBOJIOB B 0603Hat.JaeMOH <popMYJIe.
B HC4HCAeHHH MOryT HcnOAb30BaTbCSI H CHMBOAbl, He BXO.ASlWHe B ero .
HCXOAHblH (<<OCHOBHOH») aAcpaBHT; TaKHe CHMBOAbl BBO.ASlTCSI nocpeACTBOM
81
7. TeopeMbl reAeASI
onpeAeAeHHH, 3anHcaHHbiX B TepMHHax HCXOAHblX (H Bo06we paHee onpe-
AeAeHHblx) CHMBOAOB. HanpHMep, B Hawe HC4HCAeHHe MO)l(HO BBeCTH HO-
BblH CHMBOA (KoHCTaHl)') «.», 0603Ha4alOwHH COlO3 «H», onpeAeAHB ero
TaKHM 06pa30M, 4T06bl 3anHCb «p. q» BocnpHHHMaAacb KaK cOKpaweHHe
3anHCH «f"toJ (f"toJ pV f"toJ q)>>. KaKoH reAeAeBcKHH HOMep conOCTaBHTb BBeAeH-
HOMY TaKHM 06pa30M CHMBOAY? OTBeT CTaHeT cOBepweHHo 04eBHAHbIM,
KorAa Mbl 3aMeTHM, 4TO H3 Bblpa)l(eHHH, COAep)l(aWHX AonOAHHTeAbHble
(BBeAeHHble onpeAeAeHHSlMH) CHMBOAbl, 3TH HOBbie CHMBOAbl MO)l(HO HC-
KAI04HTb, 3aMeHHB HX 0603Ha4aeMblMH HMH BblpCDKeHHSlMH, COCTaBAeHHbl-
MH HCKAI04HTeAbHO H3 HCXOAHblX CHMBOAOB C 1ASI 3Toro Mbl npoCTO 4HTaeM
COOTBeTCTBYIOWHe onpeAeAeHHSI «cnpaBa HaAeBO»); M5I nOCAeAHHX )l(e
reAeAeBeKHe HOMepa Mbl CTpOHTb YMeeM. n03TOMY HaM AOCTaT04HO YCAO-
BHTbCSI C4HTaTb reAeAeBCKHM HOMepOM cpOPMYAbl «p. q» reAeAeBCKHH HO-
Mep 3KBHBaAeHTHOH eH cpOPMYAbl «f"toJ (f"toJ pV f"toJ q)>>, reAeAeBCKHM HOMepOM
«cpOPMYAbl» «2 =F 3» - HOMep 3aMeHSleMOH elO «HaCToSlweH» cpOPMYAbl
«f"toJ (ssO = sssO)>> H T. n., YCAOBHBWHCb, KOHe4HO (4TO COBceM HeTpYAHo),
06 0L1H03Ha4HOM nopSlAKe TaKHX «pacwHcppoBOK».
PaCCMOTpHM, HaKOHe , KaK)'lO-HH6Y b nOCJIe OBaTeJIb-
HOCTb <POPMYJI (MO:ry 6bITb B 1.IaCTHOM CJIyq:ae OKa-
3aTeJIbCTBOM), HanpHMep nOCJIe OBaTeJIbHOCTb
3 x (x = sy),
3 x (x = sO).
BTOpaH <poPMYJIa nOCJIe OBaTeJIbHOCTH, 1.IHTaeMaH KaK
«cyrr(eCTByer 1.IHCJIO, Henocpe CTBeHHO CJIe)JylOm:ee 3a HY-
JIeM», - BbIBO HMaH H3 nepBoH <POPMYJIbI nocpe CTBOM
nO CTaHOBKH H<PPbI 0 BMeCTO 1.IHCJIOBOH nepeMeHHoH.
* 4HTaTeAb, KOHe4HO, nOMHHT, 4TO cpopMaAbHoe AOKa3aTeAbCTBO Mbl onpe-
AeASlAH Bblwe KaK KOHe4HYIO nOCAeAOBaTeAbHOCTb cpOPMYA, KCDKAaSl H3 KO-
TOpblX eCTb AH60 aKCHOMa, AH60 BblBOAHMa H3 npeAblAYWHX cpOPMYA 3TOH
nOCAeAOBaTeAbHOCTH no npaBHAaM BblBOAa AaHHoro HC4HCAeHHSI. CorAac-
HO 3TOMY onpeAeAeHHIO npHBeAeHHaSl napa cpOPMYA AOKa3aTeAbCTBOM
82
7.1. r eAeAeBcKaSl HYMepauHSI
He SlBASleTCSI (nepBaSl ee cpopMYAa - He aKCHOMa) - 3TO AHWb «KYCOK»
AOKa3aTeAbCTBa, Ha npHMepe KOToporo Mbl CTOAb )l(e nOHSlTHO (HO AaAeKO
He TaK rpOM03AKO) nOSlCHHM HAelO HYMepaUHH, KaK H Ha ueAOM AOKa3a-
TeAbCTBe.
BbIme MbI )')Ke onpe eJIHJIH re eJIeBCKHH HOMep nepBOH
H3 3TH X <POPMYJI - MbI 0603HaQHJIH ero Tor a Qepe3 m.
TIYCTb re eJIeBCKHM HOMepOM BTOpOH <POPMYJIbI HBJUleT-
CH t.IHCJIO n. KaK H BbIme, MbI XOTHM conOCTaBHTb HameH
nOCJIe OBaTeJIbHOCTH He napy t.IHCeJI m H n, a HeKOTO-
poe e HHCTBeHHbIM 06pa30M onpe eJIeHHOe HarypaJIb-
Hoe t.IHCJIO. ,Z:(.mI 3Toro HaM OCTaTOt.IHO B3HTb B Kat.Ie-
..
CTBe TaKoro re eJIeBCKOro HOMepa t.IHCJIO, HBJUlIOII(eeCH
u
npOH3Be eHHeM CTeneHeH nepBbIX BYX npOCTbIX t.IHCeJI
CT. e. t.IHCeJI 2 H 3), npHt.IeM nepBbIH COMHO)KHTeJIb 6y-
eT BXO HTb B 3TO npoH3Be eHHe B CTeneHH, nOKa3aTeJIb
KOTOpOH paBeH re eJIeBCKOMY HOMepy nepBoH <poPMY-
JIbI, H aHaJIOrnt.IHO ,lVUI BToporo COMHO)KHTeJllI Ca TaIOKe
,lVUI TpeTbero H pyrHX, eCJIH MbI HMeeM eJIO C nOCJIe-
AOBaTeJIbHOCThIO, COCTOHII(eH 60JIee t.IeM H3 BYX <pop-
MYJI). 0603HaQHM 3TO QHCJIO t.Iepe3 k: k = 2 m X 3 n . Ta-
u u u
KOH npoCToH H KOMnaKTHbIH MeTO npHMeHHM, Ot.IeBH -
..
o, ,lVUI no.nyqeHH.H re eJIeBCKOro HOMepa npOH3BOJIb-
HOH nOCJIe OBaTeJIbHOCTH <POPMYJI. TaKHM o6pa30M, JIIO-
60e BbIpIDKeHHe HameH CHCTeMbI - 6Y b TO 3JIeMeHTap-
u
HbIH CHMBOJI, nOCJIe OBaTeJIbHOCTb CHMBOJIOB HJIH no-
u
CJIe OBaTeJIbHOCTb TaKHX nOCJIe OBaTeJIbHOCTeH - MO-
)KeT 6bITb O H03Hat.IHO 3aH)'MepOBaHO nocpe CTBOM He-
..
KOToporo re eJIeBCKOro HOMepa.
Tenepb )')Ke nOJIHaH «apH<pMeTH3anH.H» Hamero <pop-
MaJIbHOrO HCt.IHCJIeHH.H He npe CTaBHT HHKaKoro Tpy a.
TaKaH «apH<pMeTH3anIDI» nonpoc'I)' CBO HTCH K YCTa-
83
7. TeopeMbl reAeASI
HOJ3JIeHHIO HeKoToporo B3aHMHo-0 H03Hat.IHOrO COOTBeT-
CTBHH Me BbIpIDKeHHHMH, BXOAHII(HMH B HCt.IHCJIeHHe,
H HeKoTopbIM nO MHO)KeCTBOM HaTypaJIbHOrO p a.
* He BCSlKoe HaTypaAbHoe 4 HCAO eCTb reAeAeBCKHH HOMep KOHCTaHTbl, nepe-
MeHHOH HAH cpOPMYAbl; npeAOCTaBASleM 4HTaTeAIO npHAYMaTb H pa306paTb
npHMepbl Ha pa3AH4Hbie B03MO)KHble CAY4aH.
ECJIH HaM aHO KaKoe-HH6Y b BbIpIDKeHHe, MbI 6e3 Tpy a
HanHmeM ero re e.neBCKHH HOMep. 9TO, o HaKo, JIHmb
nOJ'I.ne.na. BIDKHO TO, t.ITO Kor a HaM aHO KaKOe-JIH60
HaTypaJIbHOe t.IHCJIO, TO MbI MO)KeM YCTaHOBHTb, HBJUleT-
..
CH JIH 3TO t.IHCJIO re e.neBCKHM HOMepOM, a eCJIH a -
TO MO)KeM TOt.IHO «BOCCTaHOBHTb» 0603Hat.IaeMOe 3THM
HOMepOM BbIpIDKeHHe. ECJIH aHHoe t.IHCJIO He npeBOC-
XO HT 10, TO 3TO, KaK MbI 3HaeM, npOCTO HOMep HeKO-
TOpOH KOHCTaHTbI. ECJIH )Ke aHHoe t.IHCJIO 60JIbme 10,
TO ero MO)KHO pa3JI0)KHTb, npHt.IeM e HHCTBeHHbIM 06pa-
30M (B 3TOM COCTOHT TaK Ha3bIBaeMaH OCHOBHaH TeopeMa
apHcpMeTHKH), Ha npOCTbIe COMHO)KHTeJIH. ECJIH OHO OKa-
3aJIOCb npOCTbIM, KB paTOM npOCToro HJIH Ky60M npo-
.. v v
CToro t.IHCJIa, TO 3TO - re e.neBCKHH HOMep nepeMeHHOH.
ECJIH aHHoe t.IHCJIO OKa3aJIOCb npOH3Be eHHeM CTene-
HeH nepBbIX nOCJIe OBaTeJIbHbIX npOCTbIX t.IHCeJI, TO OHO
MO)KeT (XOTH, KOHet.IHO, H He 06H3aHO) 6bITb re eJIeBCKHM
HOMepOM cpOPMYJIbI HJIH nOCJIe OBaTeJIbHOCTH <POPMYJIbI.
11 B TaKOM c.nyqae BbIpIDKeHHe, KOTOPOMY COOTBeTCTByeT
aHHbIH HOMep, MO)KeT 6bITb TOt.IHO onpe e.neHO.
CJIe H HaMet.IeHHOH nporpaMMe, MbI MO)KeM AIDI JIIO-
60ro aHHoro t.IHCJIa COBepmeHHO e HHo06pa3HbIM Me-
TO OM (<<KaK MamHHa») npOBepHTb, HBJlHeTCH JIH OHO
..
re eJIeBCKHM HOMepOM, a eCJIH a - TO KaKoro BbIpIDKe-
84
7.2. ApHcf>MeTH3auHSI MeTaMaTeMaTHKH
HIU( 2 ). TIYCTb, HanpHMep, HaM aHO 1.IHCJIO 243 000000.
Pa3JI0>KHM ero (OHO, 01.IeBHAHO, cOCTaBHoe) Ha npOCTbIe
COMHO)KHTeJIH: 243 000 000 = 64 x 243 x 15 625 =
= 2 6 X 3 5 X 56. BcnOMHHB, 1.ITO 6 eCTb re eJIeBCKHH HO-
Mep KOHCTaHTbI «0», a 5 - re eJIeBCKHH HOMep 3HaKa
«=», pHcyeM cxeMY:
6
5
6
.i
o
.i
.i
o
Teneph B HO, 1.ITO 1.IHCJIO «243 MHJIJIHOHa» eHCTBH-
TeJIbHO eCTb re eJIeBCKHH HOMep HeKoTopoH <POPMYJIbI,
a HMeHHO, <pOPMYJIhI «0 = 0» (T. e. «H)'JIb paBeH H)'JIIO»).
7.2. ApHcpMeTH3auHSI MeTaMaTeMaTH KH
CJIe m:HM marOM, KOTOphIH npO eJIaJI re eJIh, 6bI-
u
JIO 1.Ipe3BhI1.IaHHO OCTpO)'MHoe npHMeHeHHe onHcaHHoro
BhIme «KO HpOBaHIDI» (<<re eJIeBCKOH HYMepa HH»). OH
nOKa3aJI, 1.ITO Bce MeTaMaTeMarn1.IeCKHe BbICKa3bIBaHIUI
u u
o CTpYKTypHhIX CBOHCTBax BhIpIDKeHHH, BXOAHm:HX B pac-
CMaTpHBaeMoe HC1.IHCJIeHHe, MO>KHO U306pa3umb (npH-
1.IeM B3aHMHO-0 H03Ha1.IHhIM 06pa30M) B caMOM STOM HC-
1.IHCJIeHHH. B OCHOBe STOH npo eAYPbI JIe)KHT CJIe m:aH
eJI. TIOCKOJIhKy K OMY BhIpIDKeHHIO Hamero HC1.IHC-
JIeHIUI npHnHcaH HeKoTophIH (re eJIeBCKHH) HOMep, TO
2) IIoCJIe lIero Y'J(e COBceM HeTpY Ho npOBepHTb, RBJlReTCR JIH aH-
Hoe BbIpIDKeHHe cpOPMYJlOH HJIH OKa3aTeJIbCTBOM Hawero HClIHCJIeHHR
(cp. npe b ee npHMellaHHe). -l1pUM. nepea.
85
7. TeopeMbl reAeASI
K oe MeTaMaTeMamt.IeCKOe BbICKa3bIBaHHe 0 BbIpa-
>KeHIUIX HCt.IHCJIeHIUI H oTHomeHIUIX, HMelO HX MeCTO
Me HHMH, MO>KHO paccMaTpHBaTb H KaK BbICKa3bIBa-
HHe 0 cooTBeTCTBym Hx (re eJIeBcKHx) HOMepax H OT-
HomeHIUlX Me HHMH. TaKHM rryreM MeTaMaTeMaTHKa
OKa3bIBaeTCH nOJIHOCTblO «apH<pMeTH3HpOBaHHOH».
PaCCMOTpHM TaKoH nOnYJUlpHbIH npHMep. I1pH Bxo e
B 60JIbmHe )'HHBepca.nbHbIe Mara3HHbI nOKyIIaTeJUIM HHO-
r a BbI aIOT 6HJIeTHKH C HOMepaMH, Onpe eJl.HlO HMH
nopMoK aJIbHeHmero 06CJIY'KHBaHIUI nOKyIIaTeJIeH. ,lI;o-
CTaTOt.IHO 6bIBaeT nocMoTpeTb Ha 3TH HOMepa, t.IT06bI OT-
BeTHTb Ha BOnpOCbI, CKOJIbKO nOK)'I1aTeJIeH )')Ke 06CJI)')Ke-
u
HO, CKOJIbKO O>KHAaeT CBOeH Ot.Iepe H, KTO 3a KeM CTOHT,
CKOJIbKO BCerO nOK)'I1aTeJIeH 6bLnO C YTPa B Mara3HHe
H T. n. ECJIH, CKa>KeM, MHCCHC CMHT HMeeT HOMep 37,
a MHCCHC Epa)'H - HOMep 53, TO BMeCTO Toro t.IT06bI
06'bHCHHTb MHCCHC Epa)'H, t.ITO OHa OJI>KHa nponYCTHTb
Bnepe MHCCHC CMHT, OCTaTOt.IHO 06paTHTb ee BHHMa-
HHe Ha TO, t.ITO 37 MeHbme, t.IeM 53.
B MeTaMaTeMaTHKe e.no 06CTOHT B TOt.IHOCTH, KaK B 3TOM
Mara3HHe. K oe MeTaMaTeMaTHt.IeCKOe BbICKa3bIBaHHe
KO HpyeTcH Tenepb BnOJIHe 0 H03HaqHbIM 06pa30M no-
cpe CTBoM HeKoTopoH apH<pMeTHt.IeCKOH <pOPMYJIbI; OT-
HomeHIUI >Ke, HMelO He MeCTO Me MeTaMaTeMaTH-
t.IeCKHMH BbICKa3bIBaHIUIMH, H JIOrHt.IeCKHe 3aBHCHMOCTH
Me HHMH TaIOKe 0 H03Hat.IHO nepeBOAHTCH B HeKO-
TopbIe t.IHCJIOBbIe COOTHomeHIUI H 3aBHCHMOCTH Me>K-
l1.Y cooTBeTCTBym HMH apH<pMeTHt.IeCKHMH <poPMYJIaMH.
11 3 eCb, KaK H paHbme, t.IHCJIOBOe KO HpoBaHHe 06JIer-
t.IaeT paccMoTpeHHe Bcex OCTaTOt.IHO CJIO>KHbIX COOTHO-
meHHH. 113yqeHHe MeTaMaTeMaTHt.IeCKHX BonpocoB CBO-
86
7.2. ApHcf>MeTH3auHSI MeTaMaTeMaTHKH
,l(HTCH K HCCJIe,l(OBaHHIO apH<pMeTH1.IeCKHX CooTHollIeHHH
u
H CBOHCTB HeKoTopbIX 1.IHCeJI.
TIpOHJIJIIOCTpHPyeM Bce STH 06InHe 3aMe1.IaHHH O,l(HHM
SJIeMeHTapHbIM npHMepoM. B03bMeM nepByro aKcHoMY
u
HC1.IHCJIeHIDI BbIcKa3bIBaHHH, HBJI.HIOII(yIOCH, KCTaTH, aK-
CHOMOH H paccMaTpHBaeMoro CeH1.IaC JIOrHKo-apH<pMe-
TH1.IeCKOrO HC1.IHCJIeHIDI: «(p V p) ::> p». Ee re,l(eJIeBCKHH
HOMep, paBHbIH, KaK JIerKO y6e,l(HTbCH, 1.IHCJIY 2 8 X 3 11 X
X 52 X 7 112 X 11 9 X 13 3 X 17 11 MbI 0603Ha1.IHM 6YKBOH
a. PaCCMOTpHM Tenepb <POPMYJIY «(p V p)>>, re,l(eJIeBCKHH
HOMep KOTOpOH, paBHbIH 1.IHCJIY 2 8 X 3 112 X 52 X 7 112 X 11 9 ,
0603Ha1.IHM 1.Iepe3 b. C<pOPMYJIHPyeM Tenepb MeTaMa-
TeMarn1.IeCKOe yrBep eHHe, rJIaCHInee, 1.ITO <popMYJIa
«(p V p)>> eCTb Ha1.IaJIbHaH «nO,l(<popMYJIa» (T. e. 1.IaCTb
<P0PMYJIbI, caMa TaIOKe HBJUlIOInaHCH <POPMYJIOH) BbI-
6paHHOH aKCHOMbI. KaKOH apH<pMeTH1.IeCKOH <popMYJIe
paCCMaTpHBaeMOH <pOpMaJIbHOH CHCTeMbI COOTBeTCTBYeT
STO yrBep eHHe? 01.IeBHAHO, 1.ITO 60JIee KOpOTKaH <pop-
MYJIa «(pVp)>> HBJUleTCH Ha1.IaJIbHOH nO,l(<popMYJIOH 60JIee
,l(.1IHHHOH <pOPMYJIbI «(p V p) ::> p» B TOM H TOJIbKO B TOM
cn:yqae, eCJIH (re,l(eJIeBCKHH) HOMep b, COOTBeTCTByroInHH
nepBOH H3 STHX <POPMYJI, eCTb ,l(eJIHTeJIb (re,l(eJIeBCKoro)
HOMepa a, COOTBeTCTByroInerO BTOpOH <popMYJIe. B npe,l(-
nOJIO>KeHHH, 1.ITO TepMHH «,l(eJIHTeJIb» onpe,l(eJIeH He-
KOTOpbIM nO,l(XOAHInHM 06pa30M B <pOpMaJIH30BaHHOH
apH<pMeTH1.IeCKOH CHCTeMe apH<pMeTH1.IeCKOH <POPMYJIOH,
O,l(H03Ha1.IHbIM 06pa30M COOTBeTCTByroIneH ynOMHHyr0MY
BbIllie MeTaMaTeMaTH1.IeCKOMY yrBep eHHIO 0 TOM, 1.ITO
nepBaH aKCHOMa Ha1.IHHaeTCH C nO,l(<popMYJIbI «(p V p)),
HBJUleTCH <poPMYJIa «b eCTb ,l(eJIHTeJIb a». DOJIee TO-
ro, eCJIH STa nOCJIe,l(HIDI <popMYJIa HCTHHHa, T. e. eCJIH
b ,l(eHCTBHTeJIbHO HBJUleTCH ,l(eJIHTeJIeM a, TO BepHO H TO,
87
7. TeopeMbl reAeASI
qTO «(p V p)>> eCTb HaqaJIbHaJI no,n:cpoPMYJIa cpOPMYJIbI
«(p V p) :> p».
PaCCMOTpHM Tenepb nOBHHMaTeJIbHee CJIe.ZJ:YlO ee MeTa-
MaTeMaTHqeCKOe BbICKa3bIBaHHe: «nOCJIe,n:oBaTeJIbHOCTb
cpOPMYJI, HMelO aJI re,n:eJIeBCKHH HOMep x, HBJIHeTCH ,n:o-
Ka3aTeJIbCTBOM cpOPMYJIbI, HMelO eH re,n:eJIeBCKHH HO-
Mep z». BbIcKa3bIBaHHe Ko,n:HpyeTcH (H306pIDKaeTCH) no-
cpe,n:CTBoM HeKoTopoH BnOJIHe onpe,n:eJIeHHOH cpOPMYJIbI
apHcpMeTHqeCKOrO HCqHCJIeHIDI, BbIpIDKalO eH HeKOTO-
poe t.tucmo apuljjM.emut.tecKOe omHoweHue Me qHC-
JIaMH X HZ. (HeKOTOpOe npe,n:CTaBJIeHHe 0 TOM, HaCKOJIb-
KO CJIO)KHbIM HBJIHeTCH TaKOe OTHOllIeHHe, qHTaTeJIb no-
v
.nytIHT, BCnOMHHB npHBO,n:HBIlIHHCH BbIllie npHMep, B KO-
TOpOM KOHen: ,n:OKa3aTeJIbCTBa (a He Bce ,n:OKa3aTeJIb-
CTBO!) HeKOTopOH cpOPMYJIbI, HMelO eH re,n:eJIeBCKHH HO-
Mep, n, nOJIytIaJI re,n:eJIeBCKHH HOMep k = 2 m X gn.
CaMbIH 6erJIbIH aHaJIH3 npHBo,n:HT Hac K BbIBO , qTO
3,n:eCb BBO,n:HTCH BnOJIHe onpe,n:eJIeHHOe, XOTH H ,n:aJIeKO
He npOCToe, apHcpMeTHqeCKOe OTHOllIeHHe Me k (6y-
,n:eM ,n:mI npOCTOTbI CqHTaTb ero HOMepOM Bceeo ,n:OKa-
3aTeJIbCTBa) H n - re,n:eJIeBCKHM HOMepOM 3aKJIIOqeHHH
3Toro ,n:OKa3aTeJIbCTBa.) MbI 6y,n:eM 3anHCbIBaTb OTHO-
llIeHHe Me qHCJIaMH x H Z nocpe,n:CTBOM cpOPMYJIbI
«Dem(x, z)>> 3) HanOMHHalO eH HaM caMHM CBOHM 06-
JIHKOM 0 TOM MeTaMaTeMaTHqeCKOM yrBep eHHH, KO-
TOPOMY OHa COOTBeTCTByeT (a HMeHHO, 06 yrBep eHHH
«nOCJIe,n:oBaTeJIbHOCTb cpOPMYJI, HMelO aJI re,n:eJIeBCKHH
HOMep x, HBJIHeTCH ,n:OKa3aTeJIbCTBOM cpOPMYJIbI, HMelO-
eH re,n:eJIeBCKHH HOMep Z»).
3) OT aHrJI. demonstration (,n;oKa3aTeJIbCTBo). - IIpUM.. nepe6.
88
7.2. Apl / 1CpMeTH3auHSI MeTaMaTeMaTHKH
4HTaTeAb AOA)KeH TBepAo YSlCHHTb ce6e, 4TO XOTSI «Dem(x, z)>> KOAHpyeT +
HeKoTopoe MeTaMaTeMaTH4eCKoe YTBep)I(AeHHe, caMa 3Ta 3anHCb SlBASleTCSI
cpOPMYAOH apHcpMeTH4ecKoro HC4HCAeHHSI. <DopMYAa 3Ta B 60Aee npHBbl4-
HblX o603Ha4eHHSlx MO)KeT 6blTb 3anHcaHa B BHAe fez, z) = 0, rAe 6YKBa f
0603Ha4aeT HeKOTopblH AOBOAbHO-TaKH CAO)KHbIH KOMnAeKC apHcpMeTH-
4eCKHX onepaUHH HaA 4HCAaMH. OAHaKO 3Ta 60Aee npHBbl4HaSi 3anHCb
He «nOACKa3bIBaeT» cpa3Y CBoeH MeTaMaTeMaTH4eCKOH HHTepnpeTaUHH,
n04eMY Mbl H npeAn04AH 3anHCb, npHBeAeHHYIO B TeKCTe.
qHTaTe.nb Tenepb JIerKO y6e,n:HTCH B TOM, qTO MeTaMa-
TeMaTHqeCKOe yrBep eHHe, rJIaCH ee, qTO HeKOTOpaH
nOCJIe,n:OBaTe.nbHOCTb CPOPMYJI eCTb ,n:OKa3aTe.nbCTBO ,n:aH-
HOH cpOPMYJIbI, HBJIHeTCH HCTHHHbIM B TOM H TOJIbKO
.. U U
B TOM CJIyqae, eCJIH re,n:e.neBCKHH HOMep 3TOH nOCJIe,n:OBa-
Te.nbHOCTH CPOPMYJI HaxO,n:HTCH C re,n:e.neBCKHM HOMepOM
,n:aHHOH cpOPMYJIbI KaK pa3 B TOM apHcpMeTHqeCKOM OT-
HomeHHH, KOTopoe MbI 0603HaqH.1IH 3,n:eCb qepe3 «Dem».
Bo06 e, qrr06bI yrBep aTb HCTHHHOCTb HJIH JIO)KHOCTb
KaKOrO-JIH60 HHTepeCYIO ero Hac MeTaMaTeMaTHqeCKO-
ro yrnep eHIDI, HaM ,n:OCTaTOqHO pemHTb Bonpoc 0 TOM,
HaxO,l(HTCH JIH HeKOTopbIe ,n:Ba qHCJIa B OTHomeHHH, 060-
3HaqaeMOM qepe3 «Dem». Ho H 06paTHo: qT06bI y6e-
,n:HTbCH, t.ITO ,n:Ba qHCJIa HaxO,l(HTCH B Ha3BaHHOM OTHome-
HHH, ,n:OCTaTOqHQ YCTaHOBHTb HCTHHHOCTb MeTaMaTeMa-
THqeCKOrO yrnep eHIDI, «Ko,n:HpyeMoro» 3THM apHcpMe-
THt.IeCKHM OTHomeHHeM. AHaJIOrHqHO, MeTaMaTeMaTH-
qeCKOe BbICKa3bIBaHHe «nOCJIe,n:oBaTe.nbHocTb cpOPMYJI,
.. U
HMelO aH re,n:e.neBCKHH HOMep x, He HBJUIeTCH ,n:OKa3a-
Te.nbCTBOM cpOPMYJIbI, HMelO eH re,n:e.neBCKHH HOMep Z »,
KO YeTCH HeKOTopOH BnOJIHe onpe,n:e.neHHOH cpOPMYJIOH
cpOpMaJIH30BaHHOH apHcpMeTINeCKOH CHCTeMbI, HBmlIO eH-
C.H cpOpMaJIbHbIM OTpHn:aHHeM cpOPMYJIbI «Dem(x, z)>>,
T. e. cpOPMYJIOH «I'V Dem(x, z)>>.
89
7. TeopeMbl re.6.eASI
EII.J:e HeCKOJIbKO CJIOB 06 0603HaQeHIDIX, HCnOJIb3yeMbIx
B ,n:OKa3aTe.nbCTBe TeopeMbI fe,n:e.mI. HaqHeM C npHMepa.
<l>opMYJIa «3 x (x = sy)>> HMeeT re,n:e.neBcKHH HOMep m
(CM. BbIllIe, c.81), a nepeMeHHaH «y» - re,n:e.neBcKHH
HOMep 13. TIo,n:CTaBHB B 3TY cpopM)'JIY BMeCTO nepe-
MeHHoH, HMelOII.J:eH re,n:e.neBcKHH HOMep 13 (T. e. BMe-
CTO «y») IJ;HcpPY 4), 0603HaqalOII.J:YIO qHCJIO m, MbI no-
.nyqHM B pe3YJIbTaTe cpopM)'JIY «3 x (x = sm)>>, BbIpa-
malOII.J:YIO yrBep)K.lJ:eHHe, COrJIaCHO KOTOpOM)' cyrnecTByeT
TaKoe qHCJIO x, qTO 3TO x Henocpe,n:CTBeHHO CJIe.n:yeT
3a qHCJIOM m.
TIOCJIe,n:HIDI cpopMYJIa TalOKe HMeeT HeKoTopbIH re,n:e.neB-
u u
CKHH HOMep, KOTOpbIH COBceM HeTpy,n:HO BbIqHCJIHTb.
Ho BMeCTO Toro qT06bI cpaKTHqeCKH npOH3Bo,n:HTb 3TO
BbIqHCJIeHHe, MbI MomeM COBepllIeHHO 0,n:H03HaqHO oxa-
paKTepH30BaTb 3TOT HOMep qHCTO MeTaMaTeMaTHqeCKHM
06pa30M, rOBopH, qTO 3TO re,n:e.neBcKHH HOMep cpOPMYJIbI,
no.nyqaeMbIH H3 cpOPMYJIbI, HMelOII.J:eH re,n:e.neBcKHH HO-
Mep m, no,n:cTaHoBKoH BMeCTO BXOAHII.J:eH B 3TY cpOPMYJIY
nepeMeHHoH C re,n:eJIeBCKHM HOMepoM 13 IJ;HcppbI «m».
TaKaH MeTaMaTeMaTHqeCKaH xapaKTepHCTHKa 0,n:H03Haq-
HO onpe,n:e.mIeT HeKoTopoe qHCJIO, HBJUlIOII.J:eeCH HeKOTO-
pOH onpe,n:e.neHHoH cpYHKIJ;HeH OT qHCe.n m H 13, nplNeM
caMa 3Ta CPYHKIJ;IDI MomeT 6bITb BbIpIDKeHa cpe,n:cTBaMH
HallieH cpOpMaJIH30BaHHoH apHcpMeTHqeCKOH CHCTeMbI.
3HaqHT, H caMO qHCJIO MomHO BbIpa3HTb BHympu HallIe-
ro HCqHCJIeHIDI. 0603HaqHM ero qepe3 «sub(m, 13, m)>>,
..
HanOMHHaH TeM caMbIM, qTO peqb H,n:eT 0 re,n:e.neBCKOM
HOMepe cpOPMYJIbI, no.nyqeHHoH H3 cpOpM)'JIbI, HMelO-
4) [(Hcppa - 3TO qHCJIOBOH 3HaK, HJIH UMSl qHCJIa Ccp. BbIwe npH-
MeqaHHe aBTopOB Ha c. 35-36). - IIpUM.. nepea.
90
7.2. ApHcpMeTH3auHSI MeTaMaTeMaTHKH
II.J:eH re,n:e.neBcKHH HOMep m, nO,n:CTaHOBKOH 5) BMeCTO
U U ..
BXOAHII.J:eH B Hee nepeMeHHOH C re,n:e.neBCKHM HOMepoM
13 IJ;HcppbI, 0603HaqalO eH qHCJIO m. Boo6II.J:e, qepe3
«sub(y, 13, y)>> MbI 6y,n:eM 0603HaqaTb Tenepb apHcpMe-
THqecKYID CPOPMYJIY, BbIpIDKalOIQy1O BHympu apHcpMe-
THqeCKOrO HCqHcneHHH MeTaMaTeMaTHqecKYID xapaKTe-
pHCTHK)': «re,n:e.neBCKHH HOMep cpOPMYJIbI, no.nyqaeMOH
H3 cpOPMYJIbI, HMelOII(eH re,n:e.neBCKHH HOMep y, no,n:CTa-
U U U U
HOBKOH BMeCTO BXOAHII.J:eH B Hee nepeMeHHOH, HMelOII.J:eH
re,n:e.neBCKHH HOMep 13, IJ;HcpPbI, 0603HaqalOII.J:eH qHcnO
"y"». EcnH B BbIpIDKeHHe «sub(y, 13, y)>> MbI no,n:CTa-
BHM Tenepb BMeCTO «y» KaKYID-HH6y,n:b onpe,n:e.neHHyIO
IJ;HcpPY, CKameM, IJ;HcpPY; 0603HaqalOIQy1O qHcnO m, HJIH
BbIpIDKeHHe 243 000 000, TO no.nyqalOII.J:eeCH B pe3YJIb-
TaTe BbIpIDKeHHe TalOKe 6y,n:eT 0603HaqaTb HeKOTopoe
onpe,n:e.neHHoe HaTYPaJIbHOe qHcnO, HBJUIIOII(eeCH npH-
.. u u
TOM re,n:e.neBCKHM HOMepOM HeKOTopOH onpe,n:e.neHHOH
cpOPMYJIbI.
Y qHTaTeJlH He pa3 Mor B03HHKHYTb Bonpoc, noqeMY,
co6CTBeHHO, MbI rOBOpHJIH CeHqaC He npOCTO 0 «t.tuc-
Ile y», a - CTOJIb BbIqypHO H ,n:JIHHHO! - 0 «quljjpe,
0603HaqalOII(eH y». BnpOqeM, caMa cpopMa Bonpoca)')Ke
OTtlaCTH no,n:CKa3bIBaeT OTBeT. MbI Be,n:b )')Ke ynoMHHa-
JIH 0 BIDKHOM pa3JIHQHH Me nOIUlTIDIMH «qHcnO»
H «IJ;Hcppa». QHcppa - 3TO HeKOTopbIH 3HaK, T. e. BbI-
pIDKeHHe H3bIKa, KOTopoe MO)KHO 3anHCbIBaTb, CTHpaTb,
3aqepKHBaTb, nOBTopHTb H T.,n:. H T. n. lfuCllo me - 3TO
TO, UMeHe.M. (HJIH HU3BaHUe.M.,.o603Hat.teHUe.M.) Qero .HB-
JUIeTCH 0603HaQalOII.J:aH ero IJ;Hcppa; caMO no ce6e QHC-
JIO He.nb3H 3anHcaTb, crepeTb, 3aQepKHYTb, nOBTopHTb.
5) «rrO CTaHOBKa» - no-aHrJIHHCKH «substitution». - llpUM.. nepea.
91
7. TeopeMbl re.6.eASI
CKameM, Kor,n:a Mbl rOBopHM, qTO 10 - 'lUC/lO nanbneB
Ha 06eHx P)'Kax, TO Mbl XapaKTepH3yeM 3TOH cppa30H
u
HeKOTOpOe «CBOHCTBO» MHOmeCTBa HaIIIHX nanbneB -
CBOHCTBO, KOTOpOe, pa3)'MeerrCH, «nHcppOH» HHKaK He Ha-
30Beillb. Ho qHCJIO 10 MomeT 3anHCblBaThCH KaK apa6-
CKHMH nHcppaMH: «10», TaK H pHMCKHMH nHcppaMH (T. e.
nponHcHblMH JIaTHHCKHMH 6)'KBaMH) «X»; 3TH HMeHa
caMH no ce6e, KOHeqHO, pa3JIHqHbl, XOTH 0603HaqalOT
OHH O,n:HO H TO me qHCJIO. TaK BOT, Kor,n:a Mbl npOH3-
BO,n:HM no,n:CTaHOBK)' BMeCTO qHCJIOBOH nepeMeHHOH (KO-
TOpaH caMa eCTb npOCTO 3HaK, 6)'KBa), TO Mbl CTaBHM
BMeCTO o,n:Horo 3HaKa ,n:pyrOH 3HaK. Mbl He MomeM no,n:-
CTaBHTb BMeCTO 3HaKa qHCJIO - Be,n:b qHCJIO, HBJUIIOrn:e-
eCH HeKOTopblM CBOHCTBOM (HJIH, KaK HHor,n:a rOBopHT,
nOlUlTHeM), Bo06rn:e He eCTb qrrO-TO TaKoe, qTO Mom-
HO Henocpe,n:CTBeHHO HaHeCTH Ha 6)'Ma MTaK, BMeCTO
qHCJIOBOH - a .nyqIIIe CKa3aTb, qUljjPOBOi1! - nepeMeH-
HOH Mbl no,n:CTaBmIeM HMeHHO nHcpPY (HJIH nHcppOBoe
BblpIDKeHHe, CKIDKeM «sO» HJIH «7 + 5»), a He qHC-
JIO. MMeHHO n03TOMY Mbl Bblille rOBOpHJIH 0 no,n:CTa-
HOBKe qUljjPbL (0603HaqalOrn:eH qHCJIO) y, a He caMO-
ro 'lUC/la y B HHTepecyrorn:ee Hac MeTaMaTeMaTHqeCKOe
BblpIDKeHHe.
qHTaTe.nb MomeT ,n:anee nOHHTepeCOBaThCH, KaKoe me
qHCJIO 0603HaqaeTCH BblpIDKeHHeM «sub(y, 13, y)>>, ec-
JIH cpopMYJIa, HMelOInaH re,n:e.neBcKHH HOMep y, He CO-
,n:epmHT nepeMeHHOH, HMelOrn:eH re,n:e.neBCKHH HOMep 13,
T. e. nonpocTY, eCJIH cpopMYJIa He co,n:ep)KHT nepeMeHHOH
«y». CKIDKeM, sub(243 000 000, 13, 243 000000) eCTh
re,n:e.neBCKHH HOMep cpOPMYJIbl, no.nyqeHHOH H3 CPOPMY-
JIbl, HMelOrn:eH re,n:e.neBCKHH HOMep 243000 000, no,n:CTa-
92
7.2. ApHcpMeTH3auHSI MeTaMaTeMaTHKH
HOBKOH BMeCTO nepeMeHHoH «Y» n:HCPPbI6) 243000000.
BbIllie (C. 85) MbI )')Ke BblllCHH.1IH, qTO 243 000 000 -
re,ne.neBcKHH HOMep cpOPMYJIbI «0 = 0», He co,nep)Karn:eH
nepeMeHHOH «y». Ho KaKaH )Ke cpopMYJIa nOJIyqHTCH
H3 cpOPMYJIbI «0 = 0» B pe3YJIbTaTe no,nCTaHOBKH BMeCTO
He BXOAHrn:eH B Hee nepeMeHHOH «y» n:HcpPbI, 0603Ha-
qalOrn:eH qH('JIO 243 000 OOO? OTBeT OqeHb npocToH: pa3
cpopMYJIa He co,nep)KHT 3TOH nepeMeHHOH, TO H no,nCTa-
HOBKa qHCTO CPHKTHBHaH, T. e. TaKaH «no,nCTaHOBKa» He
MeHRemcpopMYJI@.HHaqerOBOpH,qH('JIO,o603HaqaeMOe
3anHCblO «sub(243 000 000, 13, 243 000 000) », eCTb ca-
MO QH('JIO 243 000 000.
3aMeTHM, HaKOHen:, QTO Bblpa)KeHHe «sub(y, 13, y)>> He
HBJUleTCH ljjOPMY/lOU. HallieH apHcpMeTHqeCKOH CHCTeMbI
B TOM CMbI('JIe, B KaKOM, HanpHMep HBJUlIOTCH cpopMYJIa-
MH Bblpa)KeHIDI «3 x (x = sy)>> H.1IH «Dem(x, z)>>, H BOT
nOQeMY. BblpIDKeHHe «0 = 0» MbI Ha3bIBaeM cpOPMYJIOH;
TaKaH 3anHCb ymBep aaem HaJIHqHe HeKOToporo OTHO-
llIeHIDI Me ,nB)'MH QH('JIaMH, TaK qTO MMeeT CMbI('JI
CTaBHTb Bonpoc, HCTHHHO H.1IH JIO)KHO 3TO yrBep)K,l];e-
HHe. AHa.n:orHQHO, Kor,na BMeCTO nepeMeHHbIX, BXOAH-
rn:HX B BblpIDKeHHe «Dem(x, Z)>>, no,nCTaBJIHIOTCH HeKO-
Topble n:HcpPbI, TO nOJIyqalOrn:eeCH BblpIDKeHMe OKa3bIBa-
eTCH 3anHCblO HeKOToporo yrBep)K,l(eHIDI (0 TOM, qTO ,nBa
QH('JIa HaXOAHTCH B HeKOTopOM OTHOllIeHHH), 0 KOTOpOM
OIDITb-TaKH HMeeT CMbI('JI CTaBHTb Bonpoc, HCTMHHO OHO
6) HanOMHHaeM, q'fO « Hcl>POH» MbI 3.n;eCb BCIO,zJ;y Ha3bIBaeM 6C1O 3U-
nuCb 'lUC/lU, a He OT.n;eJIbHbIH 3HaK TaKOH 3anHCH, KaK 06bIqHO; CKa:>KeM,
«10» eCTb Hcl>pa, 0603HaqalO aH QHCJIO 10, XOTH 06bIQHO H rOBopHT,
QTO 3TO tlHCJIO 3anHCbIBaeTCH nocpe.n;cTBoM iJ6YX Hcl>P «1» H «0». -
IIpUM.. nepe6.
93
7. TeopeMbl re.6.eASI
HJIH JIOmHO. To me caMoe MomHO CKa3aTb H 0 BbIpIDKe-
HHH «3 x (x = sy)>>.
qTO me KacaeTCH BbIpIDKeHIDI «sub(y, 13, y)>>, ,n:IDKe ec-
JIH nepecTaBHTb B Hero BMeCTO «y» KaK)'lO-HH6y,n:b KOH-
KPeTH)'IO U:HCPPY; TO OHO Bce paBHO He 6y,n:eT HHtIerO
ymaep:»caamb H no 3TOH .npHtIHHe He 6y,n:eT HH HCTHH-
HbIM, HH JIOmHbIM. BbIpIDKeHHe 3TO JIHmb 0603Ha1.laem
(HJIH J-la3blaaem) HeKOTopoe tIHCJIO, xapaKTepH3)'IOII(ee
ero KaK HeKOTOp)'IO l}jYH1Cl4UlO OT ,n:pyrHX tIHCeJI. llTaK,
BbIpIDKeHHe «Dem(x, z)>> (no,n:06HO, HanpHMep, 3anH-
CHM «y = f(x)>> HJIH «3 2 + 4 2 = 52») eCTb cpopMYJIa
H HBJUIeTCH cxeMoH (HJIH cpOpMOH) HeKOToporo ymaep-
:»caeHWl; B OTJIHQHe OT Hero 3anHCb «sub(y, 13, y)>> (no-
,n:06HO «/ (x)>> HJIH «( 7 x 5) + 8 » ) HBJIHeTCH JIHmb cxeMoH
(cpOpMOH) UMeHU HeKoToporo 1.lUC/la, HO He cpOPMYJIOH.
7.3. 1113J\o:>KeHHe AOKa3aTeAbCTB
ITepeH,n:eM, HaKOHeu:, K onHcaHHIO H,n:eH caMoro ,n:OKa-
aaTeJIbCTBa TeopeMbI re,n:e.mI. BHatI8JIe MbI ,n:a,nHM CO-
BceM npOCTOH ero Ha6poCOK, pa3,n:eJIHB ,n:OKa3aTeJIbCTBO
Ha nHTb OCHOBHbIX marOB.
ITpe e Bcero re,n:eJIb nOKa3bIBaeT (1), KaK nOCTpoHTb
apHcpMeTHtIeCK)'IO cpOPMYJIY G, npe,n:cTaBJIHIOIQyIO (<< KO-
,n:Hp)'IOIQyIO») MeTaMaTeMamqeCKOe BbICKa3bIBaHHe «cpop-
MYJIa G He,n:oKaayeMa». llHatIe rOBopH, cpopMYJIa G rJIa-
CHT 0 ce6e CQMOU, tITO OHa He,n:OKaayeMa.
ll,n:eH nOCTpoeHIDI TaKOH cpOPMYJIbI G no CYIneCTBY 3aHM-
CTBOBaHa Ha paCC)')K.n:eHIDI, npHBOAHII(erO K napa,nOKCY
94
7.3. 1I13Ao)l(eHHe AOKa3aTeAbCTB
PHllIapa. B 3TOM napMoKce, KaK MbI nOMHHM, BbIpa-
meHHIO «pHllIapoBo qHCJIO» conOCTaBJIHeTCH HeKoTopoe
qHCJIO n, nOCJIe qero paCCMaTpHBaeTCH npe,lVIOmeHHe
«n eCTb pHllIapOBO qHCJIO». B reAeJIeBCKOM me AOKa3a-
TeJIbCTBe cpopMYJIe G conOCTaBJIHeTCH HeKOTopoe qHCJIO
h, npHqeM 3TO AeJIaeTCH TaK, q'f06bI OHO COOTBeTCTBO-
BaJIO npe,lVIOmeHHIO «<I>opMYJIa, KOTOpOH conOCTaBJIeHO
qHCJIO h, HeAoKa3yeMa». Ho 3aTeM re eJIlO YAaeTCH no-
Ka3aTb (2), qTO cpopM)'JIa G AOKa3yeMa TorAa H TOJIb-
KO TorAa, KorAa AOKa3yeMo ee cpOpMaJIbHOe OTpHnaHHe
tvG. If 3TOT llIar AOKa3aTeJIbCTBa aHaJIOrHqeH COOTBeT-
CTB)'IOrn:eMY 3TOMY pacc eHHIO B napMoKce PHllIapa,
rAe AOKa3bIBaeTCH, qTO n eCTb pHllIapoBo qHCJIO B TOM
H TOJIbKO B TOM enyqae, eCJIH n He eCTb pHllIapOBO qHC-
JIO. Ho eCJIH HeKOTOpaH cpopM)'JIa H ee OTpHnaHHe AO-
Ka3yeMbI, TO apHcpMeTHqeCKOe HCqHCJIeHHe, B KOTOpOM
B03MomHbI 06a AOKa3aTeJIbCTBa, npOTHBOpeqHBO.
3HaqHT, eCJIH 3T0 HCqJfCJIeHHe HenpOTHBOpeqHBO, TO KaK G,
TaK H tv G He BbIBOAHMbI H3 aKCHOM apHcpMeTHKH. CJIe-
AOBaTeJIbHO, eCJIH apHcpMeTHKa HenpOTHBOpeqHBa, TO G
HBJIHeTCH cpOpMaJIbHO Hepa3pellIHMOH cpOpM)'JIOH. ,l(aJIee
reAeJIb AOKa3bIBaeT (3), qTO XOTH cpopMYJIa G cpOpMaJIb-
HO HeAoKa3yeMa, OHa HBJIHeTCH TeM He MeHee ucmUHHOU
apHcpMeTHqeCKOH cpOpM)'JIOH. OHa HBJIHeTCH HCTHHHOH
B TOM CMbICJIe, qTO yrBep)K,l(aeT npo K oe HaTYPaJIb-
Hoe qHCJIO, q'f0 OHO 06JIMaeT HeKoTopbIM apHcpMeTH-
u u
qeCKHM CBOHCTBOM, nplNeM CBOHCTBO 3TO TaKoro pOAa,
qTO HaJIHqHe ero Y K oro aTYPaJIbHOrO qHCJIa Mom-
u u
HO AeHCTBHTeJIbHO nOATBepAHTb nocpeAcTBoM npHMoH
npoBepKH. (4) ITOCKOJIbK)' cpopM)'JIa G, 6y.nyqH HCTHH-
HOH, HB.T.UIeTCH cpOpMaJIbHO HeAoKa3yeMoH, CHCTeMa aK-
CHOM apHcpMeTHKH HenO/1Ha. IfHbIMH CJIOBaMH, H3 aKCH-
95
7. TeopeMbl reAeASI
OM apHcpMeTHKH HeJIb3H BbIBeCTH BCe HCTHHHbIe CTpeM-
JIeHIDI apHcpMeTHKH. DOJIee TOrO, fe,n:eJIb ,n:OKa3aJI cyz.qe-
CmaeHHYlO HenOJIHory 7) apHcpMeTHKH: ,n:IDKe eCJIH npH-
COe,n:HHHTb K ee aKCHOMaTHKe HOBbIe aKCHOMbI, o6ec-
neqHBalOInHe BbIBO,n:HMOCTb HCTHHHOH cpOPMYJIbI G, Bce
paBHO H ,lVllI TaKOH nOnOJIHeHHOH (pacmHpeHHOH) CHCTe-
MbI MomHO Bcer,n:a )'Ka3aTh HCTHHHYIO, HO cpOpMaJIbHO He-
,n:OKa3yeM)'IO cpOPMYJIY. (5) B 3aKJIIOqeHHe re,n:eJIb )'Ka3aJI,
KaK nocTpoHTb apHcpMeTHqeCK)'IO cpOPMYJIY A, npe,n:CTaB-
mIlOII()'IO MeTaMaTeMaTHqeCKOe BbICKa3bIBaHHe «ApHcp-
MeTHKa HenpOTHBOpeqHBa», H ,n:OKa3aJI, qTO cpopMYJIa
«A ::> G» cpOpMaJIbHO He,n:OKa3yeMa. 113 3Toro CJIe.n:y-
eT He,n:OKa3yeMocTb H caMOH cpOPMYJIbI A. OKOHqaTeJIb-
HbIH BbIBO,n:: HenpOTHBOpeqHBOCTb apHcpMeTHKH HeJIb3H
YCTaHOBHTb nocpe,n:CTBOM paCCy)K.ZJ:eHIDI, npe,n:CTaBHMoro
B cpOpMaJIbHOM apHcpMeTHqeCKOM HCqHCJIeHHH.
ITepeii,n:eM Tenepb K 60JIee no,n:po6HOMY H3JIOmeHHIO ,n:o- .
Ka3aTeJIbCTBa TeopeMbI fe,n:eJI.H.
1. MbI)')Ke onpe,n:eJIH.1IH BbIme cpOPMYJIY «I'V Dem(x, z)),
npe,n:CTaBJUIIOII()'IO B cpOpMaJIbHOM apHcpMeTHqeCKOM HC-
qHCJIeHHH MeTaMaTeMaTH1IeCKOe BbICKa3bIBaHHe: «nOCJIe-
,n:OBaTeJIbHOCTh CPOPMYJI, HMelOInaH re,n:eJIeBCKHH HOMep x,
He HBJUIeTCH ,n:OKa3aTeJIbCTBOM cpOPMYJIbI, HMelOIneH re-
,n:eJIeBCKHH HOMep z». Tenepb MbI ,n:OCTaBHB nepe,n: cpop-
MYJIOH npHCTaBKy «\:I x», HBJUIIOII()'IOCH cpOpMaJIbHbIM
aHaJIOrOM H3bIKOBoro 060pOTa «,lVllI Bcex x» (HJIH «,lVllI
JII06oro x»), H nOJIyqHM B pe3YJlbTaTe HOB)'IO CPOPMY-
JIY «\:I X I'V Dem(x, z)), npe,n:CTaBJI.HIOII()'IO B <pOpMaJIb-
7) 9TO CBOHCTBO Ha3bIBaIOT qa e HenOnO.ll.HUMOCmblO. - llpUM..
nepe6.
96
7.3. 1I13Ao)l(eHHe AOKa3aTeAbCTB
HOM apH<pMeTHKe MeTaMaTeMaTHqeCKOe BbICKa3bIBaHHe:
«,lVUI JII06oro x nOCJIeAOBaTeJIbHOCTb <POPMYJI, HMelO-
.. U
aH reAeJIeBCKHH HOMep x, He HBmIeTCH AOKa3aTeJIb-
CTBOM <POPMYJIbl, HMelO eM reAeJIeBCKHM HOMep z». Ta-
KHM 06pa30M, 3Ta HOBaH <popMYJIa HBJllIeTCH KaK pa3
TOM <POPMYJIOM <pOpMaJIbHOrO apH<pMeTHqeCKOrO HCqHC-
JIeHIDI, KOTOpaH npeACTaBJllIeT B HeM MeTaMaTeMaTHqe-
CKoe BblCKa3blBaHHe «<popMYJIa, HMelO aH reAeJIeBCKHM
HOMep z, He,n:OKa3yeMa», H.1IH, qTO TO me: «,lVUI <POPMY-
..
JIbl C reAeJIeBCKHM HOMepOM z HeJIb3H nOCTpOHTb ,n:OKa-
3aTeJIbCTBO» .
reAMb ,n:aJIee nOKa3aJI, qTO HeKOTopblM qaCTHblH CJIY-
qaM 3TOM <POPMYJIbl HBmIeTCH <pOpMaJIbHO HeAoKa3ye-
MbIM. QTo6bl nOJIyqHTb <POPMYJI)'; Mbl 6YAeM MCXOAMTb
H3 CJIeAYI<> eM <POPMYJIbl:
X rv Dem(x, sub(y, 13, y))
(1)
9Ta <poPMYJIa, npHHMJlema a.H <pOpMaJIbHOMy apM<pMem-
qeCKoMY HCqHCJIeHHIO, npeACTaBJUIeT HeKoTopoe MeTaMa-
TeMaTHqeCKOe BblCKa3b1BaHHe. KaKoe me HMeHHo? qMTa-
TeJIb ,n:OJImeH nOMHHTb, qTO BblpameHMe «sub(y, 13, y) »
0603HaqaeT HeKoTopoe qHCJIO, KOTopoe eCTb reAeJIeBCKHH
HOMep <POPMYJIbl, nOJIyqaeMoH H3 <POPMYJIbl, HMelO eH
.. u u
re,n:eJIeBCKHH HOMep y, no,n:CTaHOBKOH BMeCTO nepeMeH-
HOM, HMelO eM re,n:eJIeBCKHM HOMep 13, (T. e. nepeMeH-
HOM y) nH<ppbl, 0603HaqalO eM qHCJIO y. OTclOAa BMAHO,
qTO <popMYJIa (1) npeACTaBJllIeT MeTaMaTeMaTMqeCKOe
BblCKa3b1BaHHe: «<popMYJIa, HMelO aH B KaqeCTBe re-
,n:eJIeBCKOro HOMepa qHCJIO sub(y, 13, y), HeAoKa3yeMa».
Ho TaK KaK <popMYJIa (1) npHHMJlemMT apM<pMeTHQe-
U .. U
CKOMY HCqHCJIeHHIO, OHa HMeeT HeKOTopblM reAeJIeBCKMH
97
7. TeopeMbl reA-eAst
HOMep, KOTOpbIH MO>KHO <paKrnqeCKH BbIqHCJIHTb. TIYCTb
STHM HOMepOM HBIDIeTCH qHCJIO n. TIo,nCTaBHM B (1) BMe-
CTO nepeMeHHoH, HMelOII(eH re,ne.neBcKHH HOMep 13 (T. e.
BMeCTO nepeMeHHoH «y»), n:H<PP 0603HaqalOIIO'IO STO
qHCJIO n. B pe3YJIbTaTe no,nCTaHOBKH MbI nOJIytlHM HeKO-
TOPYlO <poPM)'JIY, KOTOPYlO Ha30BeM (B qeCTb re,neJUl) «G»:
'V x rv Dem(x, sub(n, 13, n)).
(G)
cI>opMYJIa G H eCTb TOT qaCTHbIH CJIyqaH <pOpM)'JIbI (1),
KOTOpbIH MbI XOTe.nH nOCTpoHTb. cI>opMYJIa G npHHa.zt7Ie-
>KHT apH<pMernqeCKOM)' HCqHCJIeHHIO H ,nOJI>KHa HMeTb
HeKoTopbIH re,ne.neBcKHH HOMep. KaKoB >Ke STOT HOMep?
Herrpy,nHO nOKa3aTb, q'fO TaKHM HOMepOM 3MaeTCH qHCJIO
sub(n, 13, n). B caMOM ,ne.ne, BcnOMHHM, 'tfI'O sub(n, 13, n)
eCTb re,ne.neBcKHH HOMep <pOPM)'JIbI, no.nyqaeMoH H3 <pop-
u .. u u
M)'JIbI, HMelOII(eH re,ne.neBCKHH HOMep n, no,nCTaHOBKOH
BMeCTO nepeMeHHoH «Y» (HMelOII(eH re,ne.neBcKHH HO-
Mep 13) QH<PPbI, 0603HaqalOII(eH qHCJIO n. Ho Be,nb <pop-
M)'JIa G KaK pa3 H nOJIytleHa H3 <POPMYJIbI, HMelOII(eH
re,ne.neBcKHH HOMep n (T. e. H3 <pOpM)'JIbI (1)), no,nCTa-
HOBKOH QH<PPbI .lVUI qHCJIa n BMeCTO BXOMII(eH B <pop-
M)'JIY nepeMeHHoH y. TaKHM o6pa30M, ,neHCTBHTe.nbHO
sub(n, 13, n) eCTb re,ne.neBcKHH HOMep <POPMYJIbI G.
O,nHaKo <popMYJIa G - apU MemU1teCKWl opMYlla, KO-
TOpaH npe,nCTaBJUleT B apH<pMernqeCKOM HCqHCJIeHHH
Mame.kfamU1teCKOe BbLCKU3bLBaHue
«<popMYJIa ,,'V x rv Dem(x, sub(n, 13, n))" He,noKa3yeMa».
MO>KHO, CJIe,nOBaTe.nbHO, CKa3aTb, qTO <popM)'JIa G yrBep-
aeT CBOIO co6CTBeHH)'IO He,noKa3yeMoCTb,
98
7 .3. 3AO)l(eHllle AOKa3aTeAbCTB
2. CJIeAYIDII(HH ma KaK )')Ke rOBOpHJIOCb, COCTOHT B ,no-
Ka3aTeJIbCTBe Toro <paKTa, q'fO <popMyJIa G HBJIHeTCH
<pOpMaJIbHO He,noKa3yeMoH. OKa3aTeJIbCTBO OqeHb no-
XO>Ke Ha pacc)')K.lJ:eHHe, npHBOMII(ee K napMoKcy PH-
mapa, HO He no,nBep>KeHO TeM B03pIDKeHIDIM, KOTopbIe
BbI3bIBaeT nOCJIe,nHee.
KaK Mbl nOMHIIIM, B napaAOKCe Pillwapa cplllryplllpyeT HeKOTopoe 4111CAO n, .
CBSl3aHHoe C onpeAeAeHHblM MaTeMaTU4eCKUM Bb1CKa3b1BaHueM. B pacc}')K-
AeHIIIIII )l(e reAeASI 4111CAO n CBSl3blBaeTCSI C onpeAeAeHHoH apucpMeTu4ecKoM
cpOPMYAOM (KoTopaSi AIIIWb npeJ1CTaBAfieT MeTaMaTeMaTIII4eCKoe Bb1CKa3bl-
BaHllle). TaKIllM 06pa30M, B TeopeMe reAeASI B OTAIII4111e OT napaAOKCa PIII-
wapa IIIAeT pe4b 0 HeKoTopoM apucpMeTu4ecKoM CBOHCTBe 4111ceA (3aAa-
eTCSI Bonpoc, 06AaAaeT Alii 4111CAO sub(n, 3, n) CBOHCTBOM, Bblpa)l(aeMblM
CPOPMYAOH «V X Dem(x, sub(n, 13, n))»), a He 0 MeTaMaTeMaTU4eCKOM,
6AaroAapSi 4eMY III He B03HIIIKaeT AIIICKpeAIIITlllpYlOwero napaAOKCa Plllwa-
pa CMeweHIllSi BblCKa3blBaHillSi Ha fl3blKe aplllcpMeTIIIKIll C BblCKa3blBaHilleM
06 aplllcpMeTIIIKe.
XO,n paec)')K.lJ:eHIDI OTHOCHTeJIbHO HeCJIO>KeH. 3Maqa ero
CBO,nHTCH K TO qTo6bI ,nOKa3aTb, qTO eCJIH 6bI <poPMy-
JIa G 6bIJIa ,nOKa3yeMa, TO ee <pOpMaJIbHOe OTpHQaHHe
(T. e. <popMYJIa «rv 'V x rv Dem(x, sub(n, 13, n))» TalOKe
6bIJIO 6bI ,noKa3yeMo, H o6paTHo, eCJIH 6bI oTpHQaHHe
<POPMYJIbI G 6bIJIO ,nOKa3yeMo, TO 6bIJIa 6bI ,nOKa3yeMa
H caMa <popMYJIa G. OTCIO,na MbI nOJIyqaeM, qTO <pop-
MyJIa G ,nOKa3yeMa B TOM H TOJIbKO B TOM CJIyqae, eCJIH
,nOKa3yeMa <popMYJIa rv G.
31"0 yrBep)I(A.eHllle AOKaJaHO, crporo rOBOpSl, He CaMlllM r eAeAeM, a.6.>l<, 6. Poc- .
cepoM (1936). reAeAb)l(e nOAY4111A HeCKOAbKO 60Aee cAa6blH pe3YAbTaT, n03-
BOASIIOWIIIH, Bnp04eM, nOAY4111Tb Bce IIIHTepeCYlOwille Hac Ba)l(Hble BbIBOAbl.
BocnpOH3Be,neM BKpaTQe nepByro qaCTb paCC)')K.lJ:eHIDI
re,neJUl, COrJIaCHO KOTOpOH, eCJIH G ,nOKa3yeMa, TO H G
99
7. TeopeMbl reAeA
,nOKa3yeMa. IIYCTb G ,noKa3yeMa. Tor,na ,nOJI>KHa c e-
CTBOBaTb nOCJIe,noBaTeJIbHOCTb apH<pMeTHtleCKHX <pOPMYJI,
HBJIHIOII(aHCH ,noKa3aTeJIbCTBOM AJllI G. IIycTb re,neJIeB-
CKHH HOMep ,noKa3aTeJIbCTBa eCTb k. B TaKOM ae
Me STHM k H qHCJIOM sub(n, 13, n), HBJIHIOII(HMCH
re,neJIeBCKHM HOMepoM G, ,nOJDKHO HMeTb MecTO apH<p Me -
TINeCKOe OTHorneHHe, 0603HaqaeMOe qepe3 «Dem(x, z)>>,
T. e. «Dem(k, sub(n, 13, n)>> ,nOJI>KHa 6bITb HCTHHHOH
apH<pMeTHqeCKOH <POPMYJIOH. MO>KHO, o,nHaKO, nOKa-
3aTb, qTO STO apH<pMeTlNeCKOe OTHorneHHe 06JI aeT
u
TeM CBOHCTBOM, qTO eCJIH OHO HMeeT MeCTO AJllI KaKHX-
JIH60 ,nBYX qHCeJI, TO <popMYJIa, BbIpIDKalOII(aH STO 06-
CTOHTeJIbCTBO, HenpeMeHHO ,nOKa3yeMa. TaKHM 06pa30M,
<popMYJIa «Dem(x, sub(n, 13, n))» He TOJIbKO HCTHHHa,
HO H <pOpMaJIbHO ,nOKa3yeMa, T. e. HBJIHeTCH TeopeMOH.
Ho npaBHJIa BbIBo,na SJIeMeHTapHoH JIOrHKH n03BOJIH-
lOT HaM HeMe,lVIeHHO BbIBeCTH H3 STOH TeopeMbI <POPMY-
JIY «rv 'V x rv Dem(x, sub(n, 13, n))». TaKHM 06pa30M,
MbI BbIBeJIH H3 ,noKa3yeMocTH <POp MYJIbI G ,noKa3ye-
MOCTb ee <pOpMaJIbHOrO OTpHnaHIDI. 3 HaqHT, eCJIH Ha-
rna <pOpMaJIbHaH CHCTeMa HenpOTHBOpeqHBa, TO G B HeH
He,nOKa3yeMa.
QT06bI nOKa3aTb, qTO ,nOKa3yeMoCTb rv G BJIeqeT ,nOKa3Y-
eMOCTb G, Tpe6yeTcH aHaJIOrnqHOe, HO HeCKOJIbKO 60JIee
rpOM03,nKoe pacc eHHe, KOTopoe MbI He 6y,neM nbI-
TaTbCH 3,neCb BocnpOH3Bo,nHTb.
KaK MbI }?Ke OTMeqaJIH, eCJIH H HeKOTOpaH <popMYJIa, H ee
u
OTpHnaHHe BbIBO,nHMbI H3 HeKOTopOH CHCTeMbI aKCHOM,
TO STa CHCTeMa npOTHBOpeqHBa (HeCOBMeCTHa). IIoSTOMY
eCJIH aKCHOMbI <pOpMaJIH30BaHHoH CHCTeMbI apH<pMeTH-
KH COBMeCTHMbI, TO HH G, HH ee OTpHnaHHe He MOryr
100
7.3. V13AO)l(eHHe AOKa3aTeAbCTB
6bITb ,noKa3yeMbIMH. HHaqe rOBopH, eCJIH HalliH aKCHO-
MbI HenpOTHBOpeqHBbI, TO G <pOpMaJIbHO Hepa3peUlUMa
B TOM TOqHOM CMbICJIe, qTO HH G, HH f'V G He BbIBO,nHMbI
H3 apH<pMeTHqeCKHX aKCHOM.
3. BIDKHOCTb npe,nbI,n)'Inero 3aKJIIOqeHIDI He cpa3Y 6po-
caeTCH B rJIa3a. qTO oco6eHHoro - MOiKHO 6bIJIO 6bI
3 aTb Bonpoc - B TOM, qTO HeKOTOpaH <popMYJIa, c<pop-
MYJIHpOBaHHaH Ha apH<pMeTHqeCKOM H3bIKe, OKa3aJIaCb
Hepa3peIIIHMOH? Ho npHxo,nHTCH npH3HaTb, qTO H3 3TO-
u u
ro pe3YJIbTaTa AeHCTBHTeJIbHO BbITeKalOT qpe3BbIqaHHO
BaiKHbIe BbIBO,nbI. Bce ,ne.no B TOM, 'ITO, XOTH <popMYJIa G
H HBJI.HeTCH He,nOKa3yeMoH, MOiKHO, KaK BbIHCHHeTCH, qH-
CTO MemaMame.kfamU1.lecKUM. paCCYiK)::(eHMeM YCTaHOBMTb
ee ucmUHHocmb. HHbIMH CJIOBaMM, YAaeTC.H IIOKa3aTb,
qTO <popMYJIa G BbIpIDKaeT HeKOTopoe ( OBOJIbHO-TaKH
rpOM03,nKO BbIpa)KaeMOe, HO TeM He MeHee BIIOJIHe onpe-
,ne.neHHoe) CBOHCTBO, C Heo6xo MMOCTblO IIpMHa,z:vreiKa-
m;ee BceM HaTYPaJIbHbIM qHCJIaM (aHaJIOrMqHO, CKa)KeM,
CBOHCTBY, BbIpIDKaeMOMY ropa3,no 60JIee IIpOCTOH <pop-
MYJIOH «\I X f'V (x + 3 = 2)>>, HHTepnpeTHpyeMoM 06bIq-
HO KaK yrBepiK)::(eHHe, qTO HHKaKoe HaTYPaJIbHOe qMCJIO,
CJIO)KeHHOe C qHCJIOM 3, He ,naeT B cYMMe 2).
TIpHBe,neM 3,neCb paccylK)::(eHHe, YCTaHaBJIMBalOm;ee MC-
THHHOCTb <POPMYJIbI G. Bo-nepBbIx, B npe IIOJIO)KeHMM
HenpoTHBOpeqHBOCTH apH<pMerHKM MO)KHO AOKa3aTb, qTO
MeTaMaTeMaTHqeCKOe yrBepiK)::(eHHe
«<popMYJIa ,,\I x f'V Dem(x, sub(n, 13, n))" HeAoKa3yeMa»
I
HCTHHHO. BO-BTOPbIX, TaKoe yrBepiK)::(eHMe npe CTaBJUI-
eTCH (BbIpIDKaeTCH) B apH<pMeTHKe TOM caMoM <POp MY-
101
7. TeopeMbl reAeASI
JIOM, KOTOpaH B HeM yrroMHHaeTCH. B-TpeTbHX, MbI Bcrro-
MHHaeM, qTO HCTHHHbIM MeTaMaTHqeCKHM yrBep eHH-
.. U
JIM rrpH OC eCTBmIeMOM rrocpe,r(CTBOM re,r(eJIeBCKOH HY-
MepaQHH oT06pIDKeHHH HX B apH<pMeTHK)' COOTBeTCTBY-
lOT HCTHHHbIe )Ke apH<pMeTHqeCKHe <pOPMyJIbI. (HMeH-
HO 3TO 06CTOJlTeJIbCTBO 06YCJIOBJIHBaeT BCIO rrJIO,r(OTBOp-
HOCTb TaKoro oT06pIDKeHIDI; cHrryaQIDI 3,r(eCb COBepmeH-
U
HO Ta >Ke, q'f0 B aHaJIHTHqeCKOH reOMeTpHH, r,r(e KOOp,r(H-
HaTHoe «KO,r(HpOBaHHe» 06eCrreqHBaeT rrepeBO,r( HCTHH-
U
HbIX reOMeTpHqeCKHX BbICKa3bIBaHHH B HCTHHHbIe aJI-
re6paHqeCKHe BbICKa3bIBaHIDI.) OTCIO,r(a H BbITeKaeT, q'f0
<popMYJIa G, COOTBeTCTByroW:aH HCTHHHOMY MeTaMaTe-
MaTHqeCKOMY BbICKa3bIBaHHIO, caMa ,r(OJI>KHa 6bITb UC-
mUHHOU. CJIe,r(yeT, O,r(HaKO, ew:e pa3 rrO,r(qepKHyrb, q'f0
HCTHHHOCTb apH<pMeTHqeCKOrO BbICKa3bIBaHIDI YCTaHOB-
JIeHa HaMH OTHIO,r(b He <pOpMaJIbHbIM BbIBO,r(OM BbIpIDKa-
lOw:eM ero <POPMYJIbI H3 aKCHOM, a rrocpe,r(CTBOM HeKOTO-
poro MeTaMaTeMaTHqeCKOrO pacc eHIDI.
4. -Terrepb HaM rrpH,r(eTCJI HarrOMHHTb qHTaTeJIlO rrolUl-
THe «rrOJIHOTbI», BBe,r(eHHoe HaMH B 3aKJIIOqeHHe pa3-
,r(eJIa, rrOCBJlw:eHHoro HCqHCJIeHHIO BbICKa3bIBaHHM. MbI
Ha3BaJIH Tor,r(a CHCTeMY aKCHOM rrOJIHOM, eCJIH JII060e
HCTHHHoe rrpe,lVIO>KeHHe, BbIpa>KaeMOe Ha Jl3bIKe ,r(aH-
HOM CHCTeMbI, MO>KHO H3 HHX BbIBeCTH. B rrpOTHBHOM
CJIyqae (T. e. eCJIH He KIDK.ZJ:oe HCTHHHoe rrpe,lVIO>KeHHe,
U
BbIpa3HMoe B ,r(aHHOH CHCTeMe, BbIBO,r(HTCJI H3 ee aKCH-
OM) CHCTeMa aKCHOM «HerrOJIHa». Ho MbI TOJIbKO q'f0
KaK pa3 H YCTaHOBR7IH, qTO G eCTb HCTHHHaH apH<pMe-
THqeCKaH <popMyJIa, He BbIBO,r(HMaH H3 apH<pMeTHqeCKHX
aKCHOM, HHbIMH CJIOBaMH, CHCTeMa aKCHOM apH<pMeTHKH
HerrOJIHa (pa3YMeeTCJI, B rrpe,r(rrOJIO>KeHHH HerrpOTHBope-
102
7.3. V13AO)l(eHHe AOKa3aTeAbCTB
qHBOCTH STOH CHCTeMbl aKcHoM), 60JIee Toro, <pOpMaJIb-
HCUI apH<pMeTHKa cytqecmaeHHo HerrOJIHa: ,nIDKe eCJIH ,no-
6aBHTb K HeH <POPMYJIY G B KaqeCTBe HOBOH aKcHoMbl,
pacmHpeHHCUI CHCTeMa aKCHOM 6y,neT Bce paBHo He,nOCTa-
TOqHa .lVUI <pOpMaJIbHOrO BblBo,na acex apH<pMeTHqeCKHX
HcmH. ,l(eJIO B TOM, qTO rro OTHOmeHHIO K rrOrrOJIHeHHOH
TaKHM 06pa30M CHCTeMe aKCHOM Mbl MO)KeM rrpOBeCTH
B TOqHOCTH TO )Ke paCC)')K.l(eHHe, qTO H paHbIIIe, H Ta >Ke
u
KOHCTp)'KD:IDI ,naCT HaM HOBblH rrpHMep rrpe,lVIO)KeHIDI,
HCTHHHoro B pacmHpeHHOH apH<pMeTHqeCKOH CHCTeMe,
HO He BblBo,nHMoro H3 ee aKCHOM, H TaKoe rrpe,lVIO>KeHHe
6y,neT CHOBa BblpIDKaTbCH Hepa3pemHMOH apH<pMeTHqe-
CKOH <pOpMyJIOH. If STOT rrOHCTHHe y,nHBHTeJIbHbIH Bbl-
BO,n OCTaeTCH B CHJIe He3aBHCHMO OT Toro, CKOJIbKO pa3
Mbl HH rrpOH3Bo,nHJIH 6bl TaKoe pacmHpeHHe CHCTeMbl.
TaKHM 06pa30M, Mbl Bbl eHbl rrpH3HaTb HeKOTOpyro
u
rrpHHD:HrrHaJIbHyIO orpaHHqeHHOCTb B03MO>KHOCTeH aK-
CHOMamqeCKOrO MeTo,na. BorrpeKH, Ka3aJIOCb 6bl, caMblM
eCTeCTBeHHblM O)KH,naHIDIM, 3arrac apH<pMemqeCKHX HC-
THH OKa3blBaeTCH CTOJIb 06IIIHpHbIM, qTO HH H3 HHKaKOH
TOqHO 3a<pHKCHpOBaHHOH CHCTeMbl aKCHOM He y,naerCH
HX ace <pOpMaJIbHO BbIBeCTH.
5. Mbl rro,nOIIIJIH Terrepb K MecT)'; KOTopoe MO)KHO Ha-
u u u
3BaTb KO,nOH STO rrOpa3HTeJIbHOH HHTeJIJIeKTYaJIbHOH CHM-
<pOHHH - TBopeHIDI re,ne.IDI. OrrHcaHHble Bblme IIIarn
rr03BOJIHJIH 060CHOBaTb MeTaMareMaTHtleCKOe yrBep e-
HHe «eCJIH apH<pMeTHKa HerrpOTHBOpeqHBa, TO OHa He-
rrOJIHa». Ho re,neJIlO y,naJIOCb ,nOKa3aTb H HeqTO 60JIb-
IIIee, a HMeHHO, qTO caMO YCJIOBHOe MeTaMaTeMaTHqe-
CKoe yrBep eHHe (HMeHHo Bce yrBep eHHe B qeJlOM)
103
7. TeopeMbl reA-eAst
H306pIDKaeTCH B <pOpMaJIH30BaHHOH apH<pMernKe HeKO-
TOpOH aOKa3YeMOU <POPMYJIOH.
TIOCTpOHTb TaK)'lO 3aMeqaTeJIbH)'lO <POPMYJIY HaM 6y-
eT Tenepb COBceM HeTpy,nHO. MbI )1)Ke rOBopWIH BbI-
me (B pa3,ne.ne 5), qTO MeTaMaTeMaTHqeCKOe BbICKa3bI-
BaHHe «apH<pMeTHKa HenpoTHBOpeqHBa» 3KBHBaJIeHTHO
BbICKa3bIBaHHIO « eCTByeT xo,na 6bI o,nHa He,noKa3Y-
eMaH apH<pMeTlNeCKaH <poPMYJIa». TIOCJIe,nHee )Ke BbI-
CKa3bIBaHHe, OqeBH,nHO, npe,nCTaBmIeMCH B <pOpMaJIb-
HOM (apH<pMeTHqeCKoM) HCqHCJIeHHH CJIe,nymII(eH <pop-
u
MYJIOH:
3 y 'V x rv Dem(x, y).
(A)
cI>opMYJIa 3Ta, eCJIH BbIpa3HTb ee CJIOBeCHO" rJIaCHT: «CY-
u u
II(eCTByeT no KpaHHeH Mepe O,nHO HaTYPaJIbHOe qHCJIO y,
TaKoe q'fO ,lVllI JII06oro HaTYPaJIbHOrO x qHCJIa x H y
He HaxOMTCH Me C060H B OTHOmeHHH Dem». Ec-
JIH )Ke HHTepnpernpoBaTb <POPMYJIY KaK MeTaMaTeMaTH-
q CKOe BbICKa3bIBaHHe, TO MbI nOJIyqHM: «cym:eCTByeT
no KpaHHeH Mepe o,nHa apH<pMeTHqeCKaH <popMYJIa, ,lVllI
KOTOpOH HHKaKaH nOCJIe,nOBaTeJIbHOCTb <POPMYJI He HB-
mIeTCH ee ,nOKa3aTeJIbCTBOM». TaKHM 06pa30M, <popMYJIa
A KaK pa3 H npe,nCTaBmIeT nOCbmKY MeTaMaTeMarnqe-
CKoro yrBep eHIDI «ECJIH apH<pMernKa HenpOTHBOpe-
qHBa, TO OHa HenOJIHa». B TO )Ke BpeMH 3aKJIIOqeHHe
yrBep eHIDI «OHa (T. e. apH<pMeTHKa) HenOJIHa» He-
nocpe CTBeHHO BbITeKaeT H3 BbICKa3bIBaHIDI «HMeeTCH
HCTHHHoe apH<pMeTHqeCKOe yrBep eHHe, He HBmIlOII(e-
eCH <pOpMaJIbHO ,nOKa3yeMbIM B apH<pMeTHKe»; nOCJIe,n-
Hee )Ke BbICKa3bIBaHHe npe,nCTaB.IDleTCH B apH<pMeTHqe-
u u
CKOM HCqHCJIeHHH nocpe,nCTBOM HameH crapoH 3HaKO-
MOH - <pOPMYJIbI G. HTaK, YCJIOBHOe MeTaMaTeMarnqe-
104
7.3. V13I\O)f(eHHe .6.0Ka3aTel\bCTB
CKoe BbICKa3bIBaHHe «ECJIH apH<pMeTHKa HerrpoTHBope-
qHBa, TO OHa HerrOJIHa» rrpe,nCTaBHMO <POPMYJIOH
3 y 'V x '" Dem(x, y) :> 'V x '" Dem(x, sub(n, 13, n)),
KOTOpyro MO>KHO 6bIJIO 6bI Terrepb COKpaIQeHHO 0603Ha-
qHTb qepe3 «A :> G». HMeHHo.zvrn 3TOH <POPMYJIbI MO>K-
HO YCTaHOBHTb ee <pOpMaJIbHYID AOKa3yeMocTb, HO MbI
He 6y,neM 3,neCb rrbITaTbCH 3TO ,ne.naTb.
IIoKa>KeM JIHmb, qTO <popMYJIa A HeAoKa3yeMa. ,lJ:orry-
CTHM rrpOTHBHoe. Tor,na, rrOCKOJIbKy <popMYJIa A :> G
,nOKa3yeMa, modus ponens rr03BOJIHeT HaM 3aKJIlOqHTb,
qTO ,nOKa3yeMoH ,nOJI>KHa 6bI 6bITb H <popMYJIa G. Ho ec-
JIH Harne HCqHCJIeHHe HerrpOTHBOpeqHBO, G <pOpMaJIbHO
Hepa3pernHMa, a rrOTOMy; KOHeqHO, HeAoKa3yeMa. TaKHM
06pa30M, eCJIH apH<pMeTHKa HerrpOTHBOpeqHBa, TO <pop-
MYJIa A He,nOKa3yeMa.
qTO 3TO 03HaqaeT? <l>opMYJIa A rrpe,nCTaBJIHeT MeTaMa-
TeMaTHqeCKOe BbICKa3bIBaHHe «ApH<pMeTHKa HerrpOTH-
BOpeqHBa». 3HaqHT, eCJIH 6bI BbICKa3bIBaHHe MO>KHO 6bI-
JIO 060cHoBaTb KaKHM HH6y,nb pacc eHHeM, oT06pa-
3HMbIM B rrOCJIe,nOBaTeJIbHOCTb <POPMYJI, HBJIHIOIQYIOCH
,nOKa3aTeJIbCTBOM B apH<pMeTHqeCKOM HCqHCJIeHHH, ca-
Ma <popMYJIa A 6bIJIa 6bI ,nOKa3yeMa. Ho 3TO, KaK MbI
TOJIbKO qTO BH,neJIH, HeB03MO>KHO, eCJIH BO BCHKOM CJIY-
qae CqHTaTb, qTO apH<pMeTHKa HerrpOTHBOpeqHBa. MbI
,nOllIJIH, HaKOHen, ,no 3aKJIIOqHTeJIbHOrO aKKop,na: HaM
rrpHxo,nHTCH COrJIaCHTbCH, qTO eeJIH apH<pMeTHKa Herrpo-
THBOpeqHBa, TO HerrpOTHBOpeqHBOCTb ee He MO)KeT 6bITb
YCTaHOBJIeHa HHKaKHM MeTaMaTeMaTHqeCKHM pacc)')K-
,neHHeM, ,norrYCKaIOIQHM rrpe,nCTaBJIeHHe B apH<pMeTHqe-
CKOM <pOpMaJIH3Me.
105
7. TeopeMbl reAeASI
HMO cKa3aTb, tITO STOT 3aMetIaTeJIbHbIH pe3YJIbTaT npo-
Be,r(eHHoro re,r(eJIeM aHaJIH3a np06JIeMbI He HCKJIlOtIa-
eT, O,r(HaKO, B03MO)KHOCTH MeTaMaTeMaTHtIeCKOrO ,r(OKa-
3aTeJIbCTBa HenpOTHBOpetIHBOCTH apH<pMeTHKH. 113 Hero
CJIe,nyeT JIHmb, tITO HeB03MO)KHO TaKoe ,r(OKa3aTeJIbCTBO
HenpOTHBOpetIHBOCTH, KOTopoe MOrJIO 6bI 6bITb oTo6pa-
)KeHO (nepeBe,r(eHO) B <pOpMaJIbHOe ,r(oKa3aTeJIbCTBO, npo-
BO,r(HMoe BHYTPH caMOH <pOpMaJIbHOH apH<pMeTHKH.
* nOAO)f(eHHe 3AeCb 04eHb HanOMHHaeT TO, KOTopoe CAO)f(HAOCb B reOMeT-
pHH B CBSl3H 0 AOKa3aTeAbCTBOM HeB03MO)f(HOCTH AeAeHHSI npOH3BOAbHoro
yrAa Ha TPH 4aCTH 0 nOMOWblO UHPKYASI H AHHeHKH. AOKa3aTeAbCTBO 3TO
OTHIOAb He HCKAI04aeT B03MO)f(HOCTH npOH3BeCTH HCKOMoe AeAeHHe npH
nOMOWH KaKHX-AH60 60Aee CHAbHblX cpeACTB. VI AeHCTBHTeAbHO, ero MO)f(-
HO ocywecTBHTb, Ao6aBHB K UHPKYAIO H AHHeHKe ewe nOCTOSlHHblH 3TaAOH
A1\HHbl.
Ha caMOM ,r(eJIe MeTaMaTeMaTHtIeCKHe ,r(OKa3aTeJIbCTBa
HenpOTHBOpetIHBOCTH apH<pMeTHKH 6bIJIH nOJIyqeHbI;
nepBbIM TaKoe ,r(OKa3aTeJIbCTBO ocym:eCTBHJI npe,r(CTa-
BHTeJIb mKOJIbI IImb6epTa repxap,r( reH eH B 1936 r.,
a BnOCJIe,r(CTBHH 6bIJIO nOJIyqeHo eIne HeCKOJIbKO ,r(OKa3a-
TeJIbCTB Toro )Ke pe3YJIbTaTa. OKa3aTeJIbCTBa STH HMelOT
60JIbmyIO JIOrHtIecK}'IO eHHOCTb, 3aKJIIOtIalOIn)'IOcH xo-
TH 6bI )')Ke B TOM, tITO OHH npO,r(eMOHCTpHpOBaJIH cym:e-
CTBeHHO HOBbIe <pOpMbI MeTaMaTeMaTHtIeCKHX pacc e-
HHH H KOHCTpYK HH, a TalOKe B TOM, tITO 6JIarO,r(apH HM
BbUlCHHJIOCb, KaKHe HOBbIe BH,r(bI npaBHJI BbIBO,r(a HMO
AOl1YCTHTb, eCJIH MbI XOTHM YCTaHOBHTb HenpOTHBOpetIH-
BOCTb apH<pMeTHKH. Ho Bce nO,r(06HbIe ,r(OKa3aTeJIbCTBa
)')Ke He MOIyr 6bITb BocnpOH3Be,r(eHbI B paMKax apH<p-
MeTHtIeCKOrO HCtIHCJIeHIDI, H, nOCKOJIbK)' Bce HOBbIe npa-
BHJIa BbIBO,r(a )')Ke He HBJlHIOTCH <pHHHTHCTCKHMH, ,r(OKa-
106
7.3. 3AO)l(eHHe AOKa3aTeAbCTB
3aTeJIbCTBa HenpoTHBOpeqHBOCTH, nOJIyqeHHbIe C HX no-
MOID;blO, HHKOHM o6pa30M HeJIb3.H CqHTaTb ,nOCTH>KeHHeM
neJIH, nOCTaBJIeHHOH B rHJIb6epTOBcKOH nporpaMMe B ee
nepBOHaqaJIbHOH <pOpM)'JIHpOBKe.
8
3aKAIOLlHTeAbHbie
3aMeLlaHHSI
BbIBOAbI, K KOTOpbIM npHmeJI reAeJIb, HMelOT PHA Ba)K-
HbIX CJIeACTBHH, 6e3YCJIOBHO, He on:eHeHHbIX em;e B AOCTa-
TOtiHOH Mepe. BbIBOAbI 3TH nOKa3bIBalOT npe e Bcero,
tlTO pemeHHe 38AatiH OTbICKaHIDI .lVIH Ka OH AeAYK-
THBHOH CHCTeMbI (H B tlaCTHOCTH, ,ltJIH CHCTeMbI, B KO-
TOpOH MO)KHO 6bIJIO 6bI BbIpa3HTb BCIO COBOK)'nHOCTb
apH<pMeTHtieCKHX TeopeM) a6COJIIOTHOrO AOKa3aTeJIbCTBa
HenpOTHBOpetlHBOCTH, YAOBJIeTBOpHIOm;ero npe,ltJlO)KeH-
HbIM IRnb6epToM «<pHHHTHCTCKHM» KpHTepHHM, eCJIH
H He HBmIeTCH JIOrHtleCKH HeB03MO)KHbIM (XOTH 6bI B CH-
JIY HeKOTopOH HeOnpeAeJIeHHOCTH caMoro nOHHTIDI «CPH-
HHTHOCTH»), TO BO BC1IKOM cnyqae B BbIcmeH CTeneHH
M8JIOBepOBTHO. BbIBOAbI 3TH nOKa3bIBalOT TalOKe, qTO
HMeeTC1I 6eCKOHetiHO MHoro HCTHHHbIX apH<pMeTHqeCKHX
npeAJIO)KeHHH, KOTopbIe HeJIb3H <pOpM8JIbHO BbIBeCTH H3
u u
npoH3BOJIbHOH AaHHOH CHCTeMbI aKCHOM nocpeACTBOM He-
KOToporo TOtiHOrO nepetiIDI npaBHJI BbIBo,na. OTCIOAa
CJIe,nyeT, tlTO aKCHOMaTHtleCKHH no,nxOA K apHcpMeTHKe
HaTYP8JIbHbIX tlHCeJI, KpOMe Bcero npOtiero, He B COCTOH-
109
8. 3aKA104Jt1TeAbHble 3aMe4aHJt1S1
HHH OXBaTHTb BCIO 06JIaCTb HCTHHHbIX apHcpMeTHqeCKHX
c eHHH. OTclO,na TalOKe BbITeKaeT, qTO TO, 'tfrO MbI no-
HHMaeM no,n npOn:eCCOM MaTeMaTHqeCKOrO ,nOKa3aTeJIb-
CTBa, He CBO,nHTCH K HCnOJIb30BaHHIO aKCHOMaTHqeCKOrO
MeTo,na. cI>opManH30BaHHbIe aKCHOMaTHqeCKHe npon:e.ny-
pbI ,nOKa3aTeJIbCTB OCHOBaHbI Ha HeKOTopOM MHO)KeCTBe
BbI,neJIeHHbIX H cpHKCHpOBaHHbIX C calVi oro Haqana aK-
CHOM H npaBHJI BbIBo,na. KaK BH,nHO )')ICe H3 caMHX pac-
U ..
c eHHH, HCnOJIb30BaHHbIX B re,neJIeBCKHX ,nOKa3aTeJIb-
CTBax, H306peTaTeJIbHOCTb MaTeMaTHKOB B ,neJIe OTbICKa-
HIDI HOBbIX npaBHJI ,nOKa3aTeJIbCTBa He no,rr.naeTCH HH-
KaKHM anpHopHbIM orpaHHqeHIDIM. TaKHM 06pa30M, Co-
BepmeHHO 6e3Ha,ne:>KHO paCCqHTbIBaTb Ha TO, qTO nOIDI-
THIO y6e,nHTeJIbHOrO MaTeMaTHqeCKOrO ,nOKa3aTeJIbCTBa
MO)KHO npH,naTb pa3 HaBcer,na qeTKO OqepqeHHbIe JIOrH-
qeCKHe cpOpMbI. B CBH3H CO BceM 3THM B03HHKaeT n:e-
JIbIH pH,n HOBbIX np06JIeM, ,naneKO em;e He pemeHHbIX
H CJIHmKOM Tpy,nHbIX .lVI.H no,npo6Horo paccMoTpeHIDI
HX 3,necb - He3aBHCHMO OT Toro, MO)KHO JIH paCCqHTbI-
U U
BaTb Ha TO, qTO nOIDITIDI MaTeMaTHqeCKOH H JIOrHqeCKOH
HCTHHHOCTH MO)KHO HCqepnbIBalOm;HM 06pa30M onpe,ne-
JIHTb HJIH )Ke (MHeHHe, K KOTOPOMY CTan CKJIOIDITbCH
CQM re,neJIb) TaKoe onpe,ne.neHHe HaxO,nHTCH B KOMneTeH-
n:HH 6e30rOBOpOqHOrO cpH.JIOcocpCKoro «peaJIH3Ma» nJIa-
TOHHCTCKoro TOJIKa.
+ nAaTOHJt13M (peaAJt13M) - AOKTpJt1Ha, corAaCHO KOTOpOM MaTeMaTJt1Ka He TBO-
pJt1T Jt1 He npJt1AYMbiBaeT paCCMaTpJt1BaeMble B HeM «o6beKTbl», a OTKpblBaeT
Jt1X, nOAo6HO TOMY KaK, HanpJt1Mep, KOAYM6 OTKpblA AMepJt1KY. TaKJt1M o6pa-
30M, corAaCHO 3TOM T04Ke 3peHJt1S1, 06beKTbi AOA)KHbl B"'HeKOTopOM CMblCAe
«cywecTBOBaTb» AO Jt1X «OTKpbITJt1S1». nAaTOHJt1CTCKaSi AOKTpJt1Ha He npeA-
nOAaraeT, 4TO 06beKTbi MaTeMaTJt14eCKOro Jt1CCAeAOBaH Jt1S1 HaXOASlTCSI Me)KAy
C060M B npOCTpaHCTBeHHO-BpeMeHHbIX OTHOWeHJt1S1X. 06beKTbi 3TJt1 Cyrb
110
8. 3aKAI04HTeAbHbie 3aMe4aHHSI
OTAeAeHHble OT MaTepHaAbHblx o6oAo4eK Be4Hbie <DOpMbl, npoToTHnbl, Ha-
ceASIIOWHe oco6ble a6CTpaKTHbie Ccpepbl, AocT}'nHble AHWb V1HTeAAeKT}'.
CorAacHo TaKOIlt KOHuenUHH TpeyroAbHble HAH KpyrAble cpOpMbl cpH3H4e-
CKHX npeAMeToB, AaHHbie HaM B owyweHHSlx, caMH no ce6e BOBce He SlB-
ASiIOTCSI 06beKTaMH MaTeMaTH4eCKoro HCCAeAOBaHHSI. 3TH npOCTpaHCTBeH-
Hble cpOpMbl Cyrb AHWb HeCOBepweHHble BonAoweHHSI eAHHoro «COBep-
weHHoro» TpeyroAbHHKa HAH «COBepweHHoro» Kpyra, Be4HbIX, HeH3MeH-
HbIX, AHWb 4aCTH4HO npOSIBASIIOWHXCSI B 06AHKe MaTepHaAbHblX npeAMeTOB
H SlBASIIOWHXCSI nOAAHHHblMH 06beKTaMH paCCMOTpeHHSI MaTeMaTH4eCKOilt
MbICAH. CaM reAeAb 06HapY)KHA 6AH30CTb K TaKoro pOAa B033peHHSlM,
3aSlBASlSI, «4TO AonyweHHe... KAaCCOB H 06WHX nOHSITHIlt CTOAb )Ke 3aKOH-
HO, KaK H AonyweHHe cpH3H4eCKHX TeA... H HMelOTCSI CTOAb )Ke BblCOKHe
OCHOBaHHSI BepHTb B HX cyweCTBoBaHHe» (H3 pa60Tbi reAeASI «Russell's,
Mathematical Logic» B KHHre The Philosophy of Bertrand Russei. Evanston;
Chicago, 1944. C. 137). (AaHHaSi 3AeCb aBTopaMH xapaKTepHCTHKa «nAa-
TOHH3Ma» AOBOAbHO-TaKH nOBepxHoCTHa, a TpaAHUHOHHaSi KBaAHcpHKauHSI
reAeASI KaK nAaTOHHCTa AaAeKO He 6eccnopHa. Bnpo4eM, TeMa 3Ta AaAeKO
BblXOAHT 3a paMKH HaCTOSIWellt KHHrH. CM., HanpHMep: (/JpeHKeAb A. , 6ap-
XUMeA 11. OCHOBaHHSI TeopHH MHO)KeCTB / nep. C aHrA. M.: MHp, 1966.
rA. x.
8; 3-e H3A. M.: URSS, 2010. - npUM. nepeB.)
3aKJIlOtieHIDI, K KOTOpbIM npHme.n re,neJIb, nopo
a-
lOT, eCTeCTBeHHO, H Bonpoc, MO:>KHO JIH nOCTpOHTb BbI-
tlHCJIHTeJIbH)'lO MamHII)', cpaBHHM)'IO no CBOHM «TBOp-
tleCKHM» MaTeMaTHtieCKHM B03MO:>KHOCTHM C tle.nOBeqe-
CKHM M03rOM. COBpeMeHHbIe BbItlHCJIHTeJIbHbIe MamH-
HbI 06JIMalOT HeKOTopbIM TOtiHO cpHKCHpOBaHHbIM 3a-
naCOM KOMaH,n, KOTopbIe YMelOT BbInOJIIDITb HX 3JIeMeH-
TbI H 6JIOKH; KOMaH,nbI COOTBeTCTB)'IOT cpHKCHpOBaHHbIM
npaBHJIaM BbIBo,na HeKOTopOH <pOpMaJIH30BaHHOH aKCH-
OMaTHqeCKOH npon:eA)'pbI. TaKHM 06pa30M, MamHHa pe-
maeT 3Ma
mar 3a marOM BbInOJIHHH o,n1I)' H3 «BCTpO-
eHHbIX» B Hee 3apaHee KOMaH,n. O,nHaKO, KaK BH,nHO
.. U
H3 re,ne.neBCKOH TeopeMbI 0 HenOJIHOTe, )')ICe B 3JIeMeH-
111
8. 3aKA104Jt1TeAbHble 3aMe4aHHSI
TapHOH apHcpMeTHKe HarypaJIbHbIX qHCeJI B03HHKaer 6ec-
qHCJIeHHOe MHO)KeCTBO np06JIeM, BbIXOM
HX 3a npe-
,neJIbI B03MO)KHOCTeH JII060H KOHKpeTHOH aKCHOMaTHqe-
u
CKOH cHCTeMbI, a 3HaQHT, H He,nocrynHbIx .wrn TaKHX Ma-
mHH, CKOJIb 6bI OCTpO)'MHbIMH H CJIO)KHbIMH HH 6bLJIH
HX KOHCTp)'KI(HH H C KaKoH 6bI rpoMa,nHoH CKOpOCTblO
HH npo,neJIbIBaJIH OHH CBOH OnepaI(HH. ,l(.1lH K
OM KOH-
u
KpeTHoH 3a,naqH B npHHI(Hne MO)KHO nOCTpoHTb MamH-
H)', KOTOpOM 3Ta 3a,naQa 6blJIa 6bI no,n c
HO HeJIb-
3H c03,naTb MamHH)', npHro,nHYlO .wrn pemeHIDI JII060H
3a,naqH. IIpaB,na, H B03MO)KHOCTH qeJIOBeQeCKOrO M03ra
MOIyr OKa3aTbCH orpaHHqeHHbIMH, TaK QTO H qe.nOBeK
Tor,na CMO)KeT pemHTb He JII06)'1O 3a,naqy. Ho ,nIDKe eCJIH
3TO TaK, CTp)'KrypHbIe H CPYHKI(HOHaJIbHbIe B03MO)KHO-
CTH QeJIOBeqeCKOrO M03ra nOKa e
e HaMHoro 60JIbme
no cpaBHeHHIO C B03MO)KHOCTHMH caMbIX H30
peHHbIx
u
H3 MbICJIHMbIX nOKa MamHH, TaK qTO Henocpe,ncTBeHHoH
onaCHOCTH BbITeCHeHIDI JIIO,neH po60TaMH He BH,nHO 1) .
IIpH BceM CKa3aHHOM TeopeMY re,nemI OTHIO,nb He CJIe,ZJyer
paCI(eHHBaTb KaK HeKoe OCHOBaHHe .wrn HHTeJlJIeKTYaJIb-
Horo neCCHMH3Ma HJIH onpaB,naHIDI MHCTHqeCKHX npe,n-
CTaBJIeHHH 0 pa3)'Me. 06Hap)')KeHHe Toro cpaKTa, qTO .wrn
JII060H cpOpMaJIbHOM CHCTeMbI c
eCTB)'IOT apHcpMeTHQe-
CKHe HCTHHbI, KOTopbIe HeJIb3H B HeM cpOpMaJIbHO ,noKa-
I 3aTb, BOBce He 03HaqaeT HaJIHqIDI KaKHX-TO COBepmeHHO
Hen03HaBaeMbIX HCTHH HJIH )Ke qTO pOJIb CTpororo ,no-
1) I1pH BceM npaB
ono
06HH nOCJIe
HeH cppa3bl OHa HHKaK He CJIe-
l1.YeT H3 npe
b
ero. Bo06ll(e,
aJIeKO He HCHO, KaK pacnpocrpaHeH-
HblH Te3HC 06 orpaHH'tIeHHOCTH B03MO)KHOCTeH MO
eJlHpOBaHHH 'tIe-
JlOBeqeCKoro MblWJleHHH MO)KHO COrJlaCOBaTb C MaTepHaJIHCTH'tIeCKOH
rnnoTe30H 0 ero npHpo
e. Cp., BnpO'tleM, 3aKJIIO'tIHTeJlbHble ABa a63an;a
aBTopCKoro TeKCTa. - IIpUM. nepea.
112
8. 3aKAI04HTel\bHbie 3aMe4aHHSI
Ka3aTeJIbCTBa OTHbIHe ,nOJI)KHa 3aHHTb HeKfUI «MHCTHqe-
CKaH» HHTYH
IDI, 3aCJI)')KHBalO
aH 60JIbIlIerO ,noBepIDI,
qeM npHMeHHeMbIe HaMH cpOpMbI HHTeJlJIeKT)'aJIbHOrO
HCCJIe,nOBaHIDI. He 03HaqaeT OHO H yrBep
aeMoH HeKO-
u
TOpbIMH MbICJIHTemIMH «npHH
HnHaJIbHOH orpaHHqeH-
HOCTH qeJIOBeqeCKOrO MbIIlIJIeHIDI». 03HaqaeT OHO JIHIlIb
TO, qTO B03MO)KHOCTH HaIlIero MbIIlIJIeHIDI He CBOMT-
CH K nOJIHOCTblO cpOpMaJIH3yeMbIM npo
e.nypaM H qTO
HaM e
e npe,nCTOHT OTKpbIBaTb H H306peTaTb HOBbIe
npHH
HnbI ,nOKa3aTeJIbCTB. MbI Be,nb B
eJIH )')Ke, qTO
u
HCTHHHOCTH HeKOTopbIX MaTeMaTHqeCKHX yrBep
eHHH,
He BbIBO,nHMbIX H3 .naHHoro MHO)KeCTBa aKCHOM, MO)KHO
TeM He MeHee YCTaHOBHTb npH rrOMO
H MeTaMaTeMaTH-
qeCKHX pacc
eHHH. H yrBep)K,lJ;aTb, qTO.zvrn 060CHOBa-
HIDI TaKHX cpOpMaJIbHO He.nOKa3yeMbIx (HO YCTaHaBJIHBa-
eMbIX nocpe,nCTBOM MeTaMaTeMaTHqeCKHX pacc
eHHH)
HCTHH MO)KHO B JIyqIlIeM CJIyqae paCCqHTbIBaTb JIHIlIb
Ha HHTYH
HIO, 6bIJIO 6bI COBepIlIeHHO 6e30TBeTCTBeHHO.
KOHCTaTHpOBaHHbIe BbIIlIe OrpaHHqeHHH B03MO)KHOCTeH
BbIqHCJIHTeJIbHbIX MaIlIHH He CB
eTeJIbCTBYIOT H 0 6ec-
nOqBeHHOCTH Ha.ne
Ha 06'bHCHeHHe HBJIeHHH )KH3HH H
qeJIOBeqeCKoro MbIIlIJIeHIDI B cpH3HKO-XHMHQeCKHX Tep-
MHHax. CaMa no ce6e TeopeMa re.ne.7lll He OTBepraeT
H He no,nTBep
aeT B03MO)KHOCTH TaKoro po.na 06'bHC-
HeHHH. E.nHHCTBeHHbIH HenpeJIO)KHbIH BbIBO.n, KOTOpbIH
.. U
MbI MO)KeM c,neJIaTb H3 re,neJIeBCKOH TeopeMbI 0 HerrOJIHO-
Te, COCTOHT qTO npHpo,na H B03MO)KHOCTH qeJIOBeqeCKOrO
pa3)'Ma HeH3MepHMO TOHbIlIe H 60raqe JII060H H3 H3BeCT-
HbIX nOKa MaIlIHH. H pa60Ta caMoro re.ne.7lll HB.7llIeTCH
3aMeqaTeJIbHbIM npHMepOM 3TOH TOHKOCTH H 60raTCTBa,
,nalO
HM nOBO,n OTHIO,nb He .zvrn YHbIHHH, a, Hao6opOT,
U
.zvrn caMbIX CMeJIbIX Ha.ne
Ha CHJIY TBOpqeCKOH MbICJIH.
nOCAeCAOBHe
nepeBOA'"IHKa
KypT fe,neJIb - KpynHeHmHH CneI(HaJIHCT no MaTeMaTH-
t.JeCKOH JIOrHKe - pO,nHJICH 28 anpemI 1906 r. B EplOHHe
(HbIHe r. EpHO, qexIDI). OKOHt.JHJI BeHcKHH YHHBepcHTeT,
r,ne 3a
HTHJI ,nOKTOpCK)'IO ,nHccepTaI(HIO, 6bill ,non:eHTOM
B 1933-1938 rr. IIoCJIe aHIIIJIIOCa 3MHrpHpOBaJI B CIlIA.
C 1940 no 1963 r. fe,neJIb pa60TaeT B IIpHHCToHcKoM
HHCTH'ryTe BbIcmHX HCCJIe,noBaHHH (c 1953
- npo-
<peccop 3Toro HHCTH'ryTa). fe,neJIb - nOt.JeTHbIH ,nOKTOp
u
MeJIbCKOrO H fapBap,ncKoro yHHBepcHTeToB;, t.JJIeH Ha-
n:HOHaJIbHOH aK
eMHH Hayx CIlIA H
epHKaHcKoro
<pHJIOco<pcKoro o6
eCTBa.
B 1951
K. fe,neJIb y,noCToeH BbIcmeH HayqHoH Harp
bI
CIIIA - 9HHmTeHHoBcKOH npeMHH. B CTaTbe, nOCBH
eH-
HOH 3TOM)' C06bITHIO, ,npyrOH KpynHeHmHH MaTeMaTHK
Hamero BpeMeHH ,l()KOH <pOH HeHMaH nHCaJI 1): «BKJI
KypTa fe,nemI B COBpeMeHHyIO JIOrHKY nOHCTHHe MOHY-
MeHTaJIeH. 9TO - 60JIbme, t.JeM npoCTO MOH)'MeHT, 3TO
Bexa, pa3,nemIlO
a.H ,nBe 3nOXH... Ee3 BCHKoro npe)'Be-
1) QHTHpyeM no C60PHHKY CTaTeH «OCHoaaHWl MameMamUKU
BbI-
nyII(eHHOM)' B HbIO- HopKe B tleCTb 60-JIeTIDI K. re
e.nH (oTIY
a )Ke
B3HTbI npHBe
eHHbIe BbIwe KpaTKHe 6HorpaQ;>HtleCKHe cBe
eHHH).
115
nOCAeCAOBHe nepeBOA4HKa
JIHqeHIDI MO)KHO CKa3aTh, qrrO pa60ThI fe,nemI KOpeHHhIM
06pa30M H3MeHlUIH CaM npe,nMeT JIOrHKH KaK HaYKH».
,l(eHCTBHTeJIhHO, ,nIDKe CYXOH nepeqeHh ,nOCTH)KeHHH fe-
v
,neJUI B MaTeMaTHqeCKOH JIOrHKe nOKa3hIBaeT, tlTO HX
aBTOp no c
eCTBY 3aJIO)KHJI OCHOBhI IJ:eJIhIX pa3,neJIOB
STOR HaYKH: TeopHH Mo,neJIeH (1930 c; TaK Ha3hIBae-
MWI TeopeMa 0 nOJIHOTe Y3Koro HCqHCJIeHIDI npe,nHKaToB,
nOKa3hIBalOm;WI, rpy60 rOBopH, ,nOCTaTOtiHOCTh cpe,nCTB
«cpOpMaJIhHOR JIOrnKH» AJIH ,nOKa3aTeJIhCTBa Bcex BhI-
pIDKaeMhIx Ha ee H3hIKe HCTHHHhIX npe,ltJlO)KeHHH), KOH-
CTpYKTHBHOH JIOrHKH (1932-1933 rc; pe3YJIhTaThI 0 B03-
v
MO)KHOCTH CBe,neHIDI HeKoTophIX KJIaCCOB npe,ltJlO)KeHHH
v
KJIaCCHqeCKOH JIOrHKH K HX HHTYHI(HOHHCTCKHM aHano-
raM, nOJIO)KHBmHe HaqaJIO CHCTeMaTHtieCKOMY ynoTpe6-
v
JIeHHIO «norp)')KalOm;HX OnepaIJ:HH», n03BO.1lHIOm;HX OCY-
m;eCTB.1lHTh T-aKOe CBe,neHHe pa3JIHqHhIX JIOrnqeCKHX CH-
CTeM ,npyr K ,npyry), cpopManhHOH apHcpMeTHKH (1932-
1933 rr.; pe3YJIhTaThI 0 B03MO)KHOCTH norp)')KeHIDI KJIac-
CHqeCKOH apHcpMeTHKH B HHTYHIJ:HOHHCTCK)'IO, nOKa3hI-
BalOm;He B HeKoTOpoM CMhICJIe HenpOTHBOpeqHBOCTh nep-
BOR OTHOCHTeJIhHO BTOpOH), TeopHH anropHTMoB H pe-
K)'pCHBHhIX CPYHKIJ:HH (1934 r.; onpe,neJIeHHe nOHHTIDI
o6m;epeK)'pCHBHoH CPYHKI(HH, chIrpaBmero pemalOIIJ;yIO
v
pOJIh B YCTaHOBJIeHHH aJIrOpHTMHtleCKOH Hepa3pemHMo-
CTH PHAa BIDKHeHmHx np06JIeM MaTeMaTHKH, C O,nHOH
CTOpOHhI, H B peanH3al(HH JIOrHKO-MaTeMaTHqeCKHX 3a-
,naq Ha SJIeKTpoHHo-BhIqHCJIHTeJIhHhIX MamHHax - C ,nPY-
rOH), aKcHOMaTHtieCKOH TeopHH MHO)KeCTB (1938 c; ,no-
v
Ka3aTeJIhCTBO OTHOCHTeJIhHOH HenpoTHBOpeqHBOCTH aK-
CHOMhI BhI60pa H KOHTHHYYM-rHnOTe3hI KaHTopa OT aK-
CHOM TeopHH MHO)KeCTB, nOJIO)KHBmee HaqanO cepHH
BIDKHeRmHX pe3YJIhTaTOB 06 OTHOCHTeJIhHOH HenpOTH-
116
nOCAeCAOBHe nepeBOA4HKa
BOpeqHBOCTH H He3aBHCHMOCTH TeOpeTHKO-MHO)KeCTBeH-
HbIX npHHn:HnOB).
Ho AIDKe eCJIH 6bI Ha «Cqe'lj'» reAemI He 6blJIO HH OAHO-
v
ro H3 TaKHX 3aMeqaTeJIbHbIX ,nOCTH)KeHHH, ,nOCTaTOqHO
6blJIO 6bI OAHOH ero pa60TbI, qTo6bI HMH ee aBTopa co-
CTaBHJIO n:eJIym snoxy B HCTOpHH HaYKH. MMeHHO STOB
v
ABMn:aTHnHTHCTpaHHqHOH CTaTbe ABMn:aTHIDITHJIeTHe-
ro aBTopa H nOCBH
eHa KHH)KKa H3BeCTHoro aMepHKaH-
CKoro JIOrHKa 9. HaremI H onbITHoro nonY.7lllpH3aTopa
HaYKH ,l()K. P. HblOMeHa, nepeBeAeHHWI Ha 60JIbillHHCTBO
v
eBponeHCKHX H3bIKOB.
CpeAH AOBOJIbHO MHOrOqHCJIeHHOH K HaCTOHlI(eMY Bpe-
v v
MeHH nonY.7lllpHoH JIHTepaT)'pbI no MaTeMaTHqeCKOH JIO-
rHKe KHHra HaremI H HblOMeHa BbIAeJIHeTCH cBoeH «IJ;e-
JIeHarrpaBJIeHHOcTblO». He nbITaHCb ,naTb 061I(HH OqepK
v v
HAeH H MeTo,nOB MaTeMaTHqeCKOH JIOrHKH, aBTopbI cTpo-
HT H3JIO)KeHHe BOKpyr n:eHTpaJIbHbIX, C HX TOqKH 3peHIDI,
np06JIeMSTOHHaYKH - np06JIeM HenpoTHBOpeqHBOCTH
H nOJIHOTbI. ,l(oKa3aTeJIbCTBO Toro <paKTa, qTO AJI.H ,no-
CTaTOqHO 60raTbIx MaTeMaTHqeCKHX TeopHH Tpe60BaHIDI
STH HeCOBMeCTHMbI, H eCTb TO nopa3HTeJIbHOe OTKpbITHe
reAemI, KOTOPOMY nOCBH
eHa KHHra. He Tpe6YH OT qH-
TaTemI no cymeCTBY HHKaKHX npeABapHTeJIbHbIX n03Ha-
HHH, aBTopbI C ycnexoM 06'bHCHHIOT eM)' CymHOCTb O,nHOH
H3 caMbIX 3aMeqaTeJIbHbIX H rJIy60KHx TeopeM MaTeMa-
THKH H JIOrHKH.
CTpeMHCb K nony.7lllpHOCTH H3JIO)KeHIDI, aBTopbI ,norryc-
KalOT PHA HeTOqHOCTeH TeXHHqeCKOrO xapaKTepa. He-
MHOrOqHCJIeHHbIe HX 3aMeqaHHH <pHJIOco<pcKoro xapaK-
Tepa TalOKe npeACTaB.7llIlOTCH HeCKOJIbKO nOBepXHOCTHbI-
MH. Heo6xOAHMOCTb BOCnOJIHeHIDI TaKHX ,ne<peKTOB Ha-
117
nOCAeCAOBHe nepeBOA4HKa
p
C Tpe60BaHHeM YJIO>KHTbCH B >KeCTKO OrpaHHqeH-
HbIH 061>eM 3aCTaBHJIa nepeBO
qHKa HeCKOJIbKO cOKpa-
u
THTb TeKCT 3a CqeT HeKOTOpbIX ,lVIHHHOT, nOBTOpeHHH
H OTCTyIIJIeHHH. MecTaMH COKpam:eHHH Y
aJIOCb
06HTb-
CH n:eHOH HeKOTOpOH nepeKOMnOHOBKH MaTepHaJIa. Bce
3TH OTC1)'IL7IeHIDI OT OpHrHHaJIa Cnen:HaJIbHO HaMH He oro-
BapHBaJIHCb. Onym:eHbI TaIOKe npe
MeTHbIH )'Ka3aTeJIb
H 6H6JIHorpacpHH: qHTaTeJIb MO>KeT HaHTH
OnOJIHHTeJIb-
HbIe CCbUIKH no 3aHHTepeeoBaBlliHM ern BOnpOcaM no MO-
HorpacpHH C. K. KnHHH «BBeAeHHe B MeTaMaTeMaTHK)'»
(nep. C aHrJIHHCKOro, M.: M3
-BO HHOCTp. JIHT., 1957;
2-e H3,n. M.: URSS, 2009).
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FHeOeHKO E. B. Bece,nw 0 TeOpHH MaCCOBOrO 06CJ1Y1KHBaHHJI.
MU3ec P. BepOJITHOCTb H CTaTHCTHKa.
CmuHpoo H., l{uHH .Y: nepBble nOHJlTHJI TOnOJlOrHH.
KOJlJlZUH /0. M., CapKUCJlH A. A. n03HaKoMbTecb C TOnOJlOrHeii: Ha nO.ll.cTYnax K TOnOJlOrHH.
BOJlb6epz O. A. OCHoBHble H.lI,eH npoeKTHBHoii reOMeTPHH.
MeHHXeH (/). HeKoTopble TaiiHbI apTHCTOB-BbI'IHCJlHTeJleii.
BUIlbJIMc I(:JIC.I(. COBepweHHblii cTPaTer, HJlH BYKBapb no TeopHH CTPaTerHlJeCKHX Hrp.
IllupoKo8 ll. A. KpaTKHii OlJepK OCHOB reOMeTPHH JI06alJeBCKoro.
.' lOoUH 1(. E., /OOUH A.I(. MaTeMaTHKH H3MepJllOT CJlO}KlfOCTb.
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JIaHze B. H. <l>H3HlJeCKHe nap3.ll.0KCbI, CO
H3MbI H 3aHHMaTeJlbHble 33.l1.31JH: MexaHHKa.
MOJleKYJlJlpHaJi
H3HKa. TepMO.ll.HHaMHKa.
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H3MbI H 3aHHMaTeJlbHble 33.l1.31JH: 3J1eKTPHlJeCTBO
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H3HKa.
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H3HKe.
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FapoHep M. 3TOT npaBblii, JleBblii MHp.
C03aHo8 A. A. qeTblpexMepHaJi MO.ll.eJlb MHpa no MHHKOBCKOMY.
X80JlbCOH 0.1(. TeopHJI OTHOCHTeJlbHOCTH A. 3iiHwTeiiHa H HOBoe MHponOHHMaHHe.
BnaoUMupo8 /0. C. npOCTPaHCTBO-BpeMJI: JlBHble H CKpblTble pa3MepHOCTH.
llepeJlbMaH H. H. 3aHHMaTeJlbHaJi aCTPOHOMHJI.
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