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.

axiomatic sets 30 (making a subset 2)

By axiom of separation, we are able to make a new little set whose elements are chosen from a given set.
\[ \forall x\exists y\forall y[z\in y\leftrightarrow z\in x\wedge P(x)]  \]

For example, if $P(x)$  is $z\in w$ ,then
\[ \left\{z\in x:z\in w \right\}=y=x\cap w .  \]

However, we are only able to make a few new sets by using this axiom.
We want to make a bigger set freely.

A.Fraenkel offered an alternative axiom "axiom of replacement".
\[  \forall x\exists y \forall z[z\in y\leftrightarrow \exists w[w\in x \wedge \phi(w,z)]]  \]
 ,where $\phi$ is a formula or a statement of properties or a function.

That is to say, if
\[  \forall x\in a,\quad \exists y ,\quad \phi(x,y) \]
 is satisfied, then
\[  z=\left\{ y:\phi(x,y),x\in a \right\} \]
becomes a set.

More simple, using a function $\phi(x)=y$ ,
it is possible for us to make a new set $(y\in) z$ by a given set $a$ whose elements are $x$.

A function (this is a set, too) for a given set can make a new set. it is very strong extension.

Of course, axiom of separation can be proved by axiom of replacement easily.