Editing Zermelo–Fraenkel set theory (section)

=== Axiom of infinity ===
{{Main|Axiom of infinity}}

{| class="floatright" style="background-color: #f8f9fa; border: 1px solid #a2a9b1; margin: 0.5em 0 0.5em 1em; padding: 0.2em; color:black;" <!-- Black is *not* the default color for text (yes, really!) -->
|+ First several von Neumann ordinals
|-
! scope="row" | 0
| = || {}
| = || ∅
|-
! scope="row" | 1
| = || {0}
| = || {∅}
|-
! scope="row" | 2
| = || {0,1}
| = || {∅,{∅}}
|-
! scope="row" | 3
| = || {0,1,2}
| = || {∅,{∅},{∅,{∅}}}
|-
! scope="row" | 4
| = || {0,1,2,3}
| = || {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
|}

Let <math>S(w)</math> abbreviate <math>w \cup \{w\},</math> where <math> w </math> is some set. (We can see that <math>\{w\}</math> is a valid set by applying the axiom of pairing with <math>x = y = w</math> so that the set {{mvar|z}} is <math>\{w\}</math>). Then there exists a set {{mvar|X}} such that the empty set <math>\varnothing</math>, defined axiomatically, is a member of {{mvar|X}} and, whenever a set {{mvar|y}} is a member of {{mvar|X}} then <math>S(y)</math> is also a member of {{mvar|X}}.

<div style="margin-left:1.6em;"><math>\exists X \left [\exists e (\forall z \, \neg (z \in e) \land e \in X) \land \forall y (y \in X \Rightarrow S(y)  \in X)\right].</math></div>

or in modern notation:
<math>\exists X \left [\varnothing \in X \land \forall y (y \in X \Rightarrow S(y)  \in X)\right].</math>

More colloquially, there exists a set {{mvar|X}} having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set {{mvar|X}} satisfying the axiom of infinity is the [[von Neumann ordinal]] {{mvar|&omega;}} which can also be thought of as the set of [[natural numbers]] <math>\mathbb{N}.</math>