Set-theoretic models Peano axioms

the peano axioms can derived set theoretic constructions of natural numbers , axioms of set theory such zf. standard construction of naturals, due john von neumann, starts definition of 0 empty set, ∅, , operator s on sets defined as:

s(a) = ∪ { }.

the set of natural numbers n defined intersection of sets closed under s contain empty set. each natural number equal (as set) set of natural numbers less it:

0



=
∅




1



=
s
(
0
)
=
s
(
∅
)
=
∅
∪
{
∅
}
=
{
∅
}
=
{
0
}




2



=
s
(
1
)
=
s
(
{
0
}
)
=
{
0
}
∪
{
{
0
}
}
=
{
0
,
{
0
}
}
=
{
0
,
1
}




3



=
s
(
2
)
=
s
(
{
0
,
1
}
)
=
{
0
,
1
}
∪
{
{
0
,
1
}
}
=
{
0
,
1
,
{
0
,
1
}
}
=
{
0
,
1
,
2
}






{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}

and on. set n 0 , successor function s : n → n satisfies peano axioms.

peano arithmetic equiconsistent several weak systems of set theory. 1 such system zfc axiom of infinity replaced negation. such system consists of general set theory (extensionality, existence of empty set, , axiom of adjunction), augmented axiom schema stating property holds empty set , holds of adjunction whenever holds of adjunct must hold sets.