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10
1.Basicalgebraicpropertiesofintegers
whereΦ(n)isanyformulaofthetheory;orthisaxiomcanhaveanonele-
mentary(secondorder)form(whereweacceptvariablesrunningoverthe
setofsubsetsofN):
^
[(0∈X)∧(n∈X⇒s(n)∈X)]⇒(X=N).
X⊂N
Thesecondwayrequireshavingsomesettheory.
However,theaxiomaticwayofintroducingnaturalnumbersneglects
intuitivesourcesofthesenumbers.Hencealsotheconstructionofamodel
ofnaturalnumbersascomposedofthecardinalitiesoffinitesetsinsomeset
theoryhasoftenbeenconsidered.
Anequivalentwayofconstructingamodelofnumbersinsettheoryhas
beenproposedbyJohnvonNeumann;itconsistsindefiningaspecificset
ofanyfixedfinitecardinality.First,welet0betheemptyset,andwhen
wehavealreadychosenafinitesetX,thenwetakethesuccessorofXto
beX∪{X}.Forexample,thenumber1isrepresentedbytheone-element
set∅∪{∅}.
Thetwowaysleadtotwo“theoriesofnumbers”,withdifferentsetsof
theorems.Intheaxiomaticcase,thetheoremsaretheresultsthatcanbe
derivedfromtheaxioms;intheset-theoreticcase,thetheoremsarethe
resultsthataretrueinthemodel.
Itisnotpossibletoderivealltheresultstrueinagivenmodelofthenat-
uralnumbersfromanygiven“effectivelydefined”familyofaxioms(K.Gödel,
1936).Thisfollowsfromthefactthatintheaxiomaticnumbertheoryone
mayprescribetoanyofitsformulasanaturalnumberandthentreatthe
reasoningstepsasfunctionsonthenaturalnumbers.
Thisleadstothepossibilityofstatingasentencesayingaboutitselfthat
itisnotaconsequenceofaxioms.Thenneitherthissentence,calledthe
“Gödelsentence”,noritsnegation,isaconsequenceoftheaxioms.
Itisveryinterestingthatasentencewhichcannotbeprovedwithin
anyeffectivelygivenaxiomaticnumbertheorycanalwaysbefoundamong
sentencesconcerningexistenceofsolutionsinZofanequation
f(x1,...,xn)=0,
(∗)
wheref(x1,...,xn)∈Z[x1,...,xn].ThisresultprovedbyYuriiMatyasevich
(1970)answeredHilbert’sTenthProblem.Itisknownthatsuchapolynomial
f(x1,...,xn)cannotbeofdegree1or2;itisnotknownifitcanbeof
degree3.
Anyway,numbertheoryistoorichtobeaxiomatizable.However,sofar,
itseemsthatthisambiguityofthenotionoftruthofsentencesdescribing
propertiesofthenaturalnumbersdoesnotconcernthepropertiesthatmath-
ematiciansconsidertobeinterestingwithinnumbertheory.Nevertheless,
