Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
1.Basicalgebraicpropertiesofintegers
1.Theoriginandfoundationsofnaturalnumbers
Formanycenturiesmathematiciansconsiderednaturalnumbersasgiven(by
God)anddidnotbotherabouttheirorigin.However,inthelate19thand
early20thcentury,itbecameevidentthatoneshouldgiveamoreprecise
meaningtomathematicalobjectsaswellastomathematicalreasoning,and
peoplebelievedthatafoundationofmathematicaltheoriescanbegivenby
determiningtheiraxioms(andrulesofreasoning).
Aftersometrials,asetofsuchaxiomsofthetheoryofnaturalnumbers
proposedbyPeanowerecommonlyaccepted.Theseaxiomscanbedescribed
inthefollowingway.First,onefixestheprimitivenotionsof
♦anaturalnumberandthesetNofnaturalnumbers,
thenumber0∈N,and
thesuccessorfunctions:N→N
andoneadmitsthefollowingaxiomsdescribingthepropertiesoftheseno-
tions:
1.A
(s(n)=s(m)⇒n=m),
2.A
3.InductionAxiom.
n,m
n
(s(n)/=0)and
TheInductionAxiomcaneitherhaveanelementary,i.e.,firstorderform
(whereweacceptonlyvariablesrunningoverN)andisgivenasascheme
(sequence)ofexpressions
[Φ(0)∧^
(Φ(n)⇒Φ(s(n)))]⇒^
Φ(m),
n
m
