axiomatic sets 33 (regularity 2)

Axiom of regularity is also called axiom of foundation.
\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]

This axiom requires that even infinite sets are bounded
and all collections of things is not necessarily a set.

Example(1); $A=\left\{ x: x\notin x \right\}$  is not a set.

$A$  is a collection of a set which does not contain itself.
(you are able to consider that $A$  is a very huge collection. )

If you accept that $A$  is a set, then a contradiction occurs.

We will give you a question.
Which do you prefer, $A\in A$  or $A\notin A$ ?

This is Russell's paradox.

However, using axiom of regularity we have to recognize $A$  is not a set.
If $A\in A$ is accepted, as there exists the sequence $A\ni A\ni A\ni\cdots$ ,
$A$ violates the axiom.

If $A\notin A$ is accepted, $\left\{ x: x\notin x \right\}$ is not satisfied.


Example(2); The collection $A$ of all sets is not a set.

$A$ is the largest collection in all sets.

However, as $A\in A$ , there is a sequence $A\ni A\ni A\ni\cdots$ .
Therefore, by axiom of regularity $A$ is not a set.

The collections which are not set are called classes.