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.