Every Set in Von Neumann Universe
Theorem
Let $S$ be a small class.
Then $S$ is well-founded.
The connection between "well-founded" and "in von Neumann universe".
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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 $\displaystyle \bigcup_{a \mathop \in S} F \left({a}\right)$.
... and do what with it?
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Take any $a \in S$.
| \(\ds a\) | \(\in\) | \(\ds V \left({F \left({a}\right)}\right)\) | Definition of $F$ | |||||||||||
| \(\ds \implies \ \ \) | \(\ds a\) | \(\in\) | \(\ds V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) | Set is Subset of Union: Family of Sets and Von Neumann Hierarchy Comparison | ||||||||||
| \(\ds \implies \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) | Definition of Subset | ||||||||||
| \(\ds \implies \ \ \) | \(\ds S\) | \(\in\) | \(\ds \mathcal P \left({ V\left({ \bigcup_{x \mathop \in S} F\left({x}\right) }\right) }\right)\) | Definition of Power Set |
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: $\S 9.13$