Every Set in Von Neumann Universe

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: $\S 9.13$