# Every Set in Von Neumann Universe

Jump to navigation
Jump to search

## Theorem

Let $S$ be a small class.

Then $S$ is well-founded.

## Proof

*This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.*

*If you see any proofs that link to this page, please insert this template at the top.*

*If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.*

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

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

That is, $a \in \map V x$ for some $x$.

Let:

- $\ds \map F a = \bigcap \set {x \in \On : a \in \map V x}$

Take $\ds \bigcup_{a \mathop \in S} \map F a$.

Take any $a \in S$.

\(\ds a\) | \(\in\) | \(\ds \map V {\map F a}\) | Definition of $F$ | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds a\) | \(\in\) | \(\ds \map V {\bigcup_{x \mathop \in S} \map F x}\) | Set is Subset of Union: Family of Sets and Von Neumann Hierarchy Comparison | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds \map V {\bigcup_{x \mathop \in S} \map F x}\) | Definition of Subset | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds S\) | \(\in\) | \(\ds \powerset {\map V {\bigcup_{x \mathop \in S} \map F x} }\) | Definition of Power Set |

Therefore:

- $\ds S \in \map V {\bigcup_{x \mathop \in S} \map F x + 1}$

and $S \in \map V x$ for some ordinal $x$.

$\blacksquare$

## Sources

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