> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/set-theory-axioms/axiom-of-infinity.md).

# Axiom of infinity

**Axiom of infinity**

The union of a set $$n$$ with a set containing the set $$n$$ is denoted by: $$n\cup {n} = {n, {n}}$$

If we abbreviate this particular union operation using the prime symbol, we get the notation: $$n' = n\cup {n} = {n, {n}}$$

If $$n = \varnothing$$ then

$$
n' = \varnothing'
\= \varnothing \cup { \varnothing }
\= { \varnothing, { \varnothing }}
$$

By applying the *axiom of pairing*, $$\forall x\forall y\ \exists z\ (x\in z\land y\in z)$$, we see that a set $${n}$$ is a valid set, because the axiom of pairing states that if $$x$$ and $$y$$ are sets then there is a set, $$z$$, that contains them as its elements.

> then there exists a set X such that the empty set is a member of $$X$$ and, whenever a set $$y$$ is a member of $$X$$, then $$y' = y\cup {y}$$ is also a member of $$X$$.

$$\exists X \[\varnothing\in X \land \forall y(y\in X\to y' \in X)]$$

Less formally, it states that there exists a set 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 X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers $$\mathbb{N}$$.

## Von Neumann ordinals

```
0 = { } = ∅
1 = { 0 } = {∅}
2 = { 0, 1 } = { ∅, {∅} }
3 = { 0, 1, 2 } = { ∅, {∅} , {∅, {∅)) }
4 = { 0, 1, 2, 3 } = { ∅, {∅}, {Ø, {Ø)), {Ø, {Ø}, {Ø, {Ø))} }
etc.
```