axiomatic sets 31 (infinite sets)

In number system, We have already used the axiom of infinity.

\[ \exists x[0\in x\wedge \forall y[y\in x\rightarrow (y\cup \left\{ y\right\})\in x] ]  \]

 ,where $(y\cup \left\{ y\right\})$ is called the successor set of $y$ .

In axiomatic set theory, an infinite set is the set $x$ such that if an element $y$ is in $x$ , then the successor set of $y$ is also in $x$ .

Namely, if $0\in x$ , then the successor set $1$ of $0$ is in $x$ .
As $1\in x$ , the successor set $2$ of $1$ is in $x$ , $\cdots$ and so on.

This infinite set has no maximum number in the set.

You must have seen by this axiom there is $\mathbb{N}$ as the base of numbers.

This axiom asserts there are sets whose elements are infinite.