# Every Set in Von Neumann Universe

## Theorem

Let $S$ be a small class.

Then $S$ is well-founded.

## Proof

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The proof shall proceed by Epsilon Induction on $S$.

Suppose that all the elements $a \in S$ are well-founded.

That is, $a \in V \left({x}\right)$ for some $x$.

Let:

- $\displaystyle F \left({a}\right) = \bigcap \left\{{x \in \operatorname{On} : a \in V \left({x}\right)}\right\}$

Take $\displaystyle \bigcup_{a \mathop \in S} F \left({a}\right)$.

Take any $a \in S$.

\(\displaystyle a\) | \(\in\) | \(\displaystyle V \left({F \left({a}\right)}\right)\) | $\quad$ Definition of $F$ | $\quad$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle a\) | \(\in\) | \(\displaystyle V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) | $\quad$ Set is Subset of Union: Family of Sets and Von Neumann Hierarchy Comparison | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle S\) | \(\subseteq\) | \(\displaystyle V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) | $\quad$ Definition of Subset | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle S\) | \(\in\) | \(\displaystyle \mathcal P \left({ V\left({ \bigcup_{x \mathop \in S} F\left({x}\right) }\right) }\right)\) | $\quad$ Definition of Power Set | $\quad$ |

Therefore:

- $\displaystyle S \in V \left({ \bigcup_{x \mathop \in S} F \left({x}\right) + 1}\right)$

and $S \in V \left({x}\right)$ for some ordinal $x$.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 9.13$