# Ordinal is not Element of Itself/Proof 3

## Theorem

Let $x$ be an ordinal.

Then:

- $x \notin x$

## Proof

Let $\On$ denote the class of all ordinals.

From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$ is superinductive.

Hence we can use the Principle of Superinduction.

By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$.

We identify the natural number $0$ via the von Neumann construction of the natural numbers as:

- $0 := \O$

From Empty Set is Ordinary:

- $\O \notin \O$

and so:

- $0 \notin 0$

$\Box$

Let $x$ be an ordinal for which it has been established that:

- $x \notin x$

Then by Successor Set of Ordinary Transitive Set is Ordinaryā€ˇ:

- $x^+ \notin x^+$

$\Box$

It remains to be shown that if $C$ is a chain of ordinals all of which are ordinary, then $\bigcup C$ is ordinary.

Let $C$ be a chain of ordinals.

From Union of Set of Ordinals is Ordinal, $\bigcup C$ is itself an ordinal.

Let $\alpha = \bigcup C$.

Aiming for a contradiction, suppose $\bigcup C$ is not ordinary.

That is, suppose that $\alpha \in \alpha$.

Then:

- $\alpha \in \bigcup C$

Hence for some $\beta \in C$:

- $\alpha \in \beta$

But:

- $\beta \in C$

and so:

- $\beta \subseteq \bigcup C$

That is:

- $\beta \subseteq \alpha$

Because $\alpha \in \beta$ and Ordinal is Transitive:

- $\alpha \subseteq \beta$

Hence by definition of set equality:

- $\alpha = \beta$

That is:

- $\beta \in \beta$

Thus some element of $C$, that is, $\beta$, is an element of itself.

That is, not all elements of $C$ are ordinary.

This contradicts our assertion that all elements of $C$ are ordinary.

Hence by Proof by Contradiction we have that if $C$ is a chain of ordinals all of which are ordinary, then $\bigcup C$ is ordinary:

- $\bigcup C \notin \bigcup C$

$\Box$

The result then follows by the Principle of Superinduction.

$\blacksquare$

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.8$