Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy

Theorem

Let $x$ be an ordinal.

Let $S$ be a small class.

Let $\map V x$ denote the von Neumann hierarchy on the ordinal $x$.

Then $x$ is a subset of the rank of $S$ if and only if $S \notin \map V x$.

Proof

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Necessary Condition

Let $x \subseteq \map {\operatorname{rank} } S$.

Then by Von Neumann Hierarchy Comparison:

$S \in \map V x \implies S \in \map V {\map {\operatorname{rank} } S}$

But by Ordinal Equal to Rank:

$S \notin \map V {\map {\operatorname{rank} } S}$

By contraposition:

$S \notin \map V x$

$\Box$

Sufficient Condition

Let $S \notin \map V x$.

Then:

\(\ds S\)

\(\in\)

\(\ds \map V {\map {\operatorname{rank} } S + 1}\)

Ordinal Equal to Rank

\(\ds \leadsto \ \ \)

\(\ds \map V {\map {\operatorname{rank} } S + 1}\)

\(\nsubseteq\)

\(\ds \map V x\)

Rule of Transposition

\(\ds \leadsto \ \ \)

\(\ds \map {\operatorname{rank} } S + 1\)

\(\nsubseteq\)

\(\ds x\)

Von Neumann Hierarchy Comparison

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds \map {\operatorname{rank} } S + 1\)

Transitive Set is Proper Subset of Ordinal iff Element of Ordinal and Ordinal Membership is Trichotomy

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\subseteq\)

\(\ds \map {\operatorname{rank} } S\)

Definition of Successor Set

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.15 \ (3)$