Editing Peano axioms (section)

==== Set-theoretic models ====
{{Main|Set-theoretic definition of natural numbers}}
The Peano axioms can be derived from [[set theory|set theoretic]] constructions of the [[natural number]]s and axioms of set theory such as [[Zermelo–Fraenkel set theory|ZF]].<ref>{{harvnb|Suppes|1960}}, {{harvnb|Hatcher|2014}}</ref> The standard construction of the naturals, due to [[John von Neumann]], starts from a definition of 0 as the empty set, ∅, and an operator ''s'' on sets defined as:

: <math>s(a) = a \cup \{a\}</math>

The set of natural numbers '''N''' is defined as the intersection of all sets [[closure (mathematics)|closed]] under ''s'' that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:

: <math>\begin{align}

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{align}</math>
and so on. The set '''N''' together with 0 and the [[successor function]] {{nowrap|''s'' : '''N''' → '''N'''}} satisfies the Peano axioms.

Peano arithmetic is [[equiconsistent]] with several weak systems of set theory.{{sfn|Tarski|Givant|1987|loc=Section 7.6}} One such system is ZFC with the [[axiom of infinity]] replaced by its negation. Another such system consists of [[general set theory]] ([[Axiom of extensionality|extensionality]], existence of the [[empty set]], and the [[general set theory|axiom of adjunction]]), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.