Von Neumann Hierarchy Comparison

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Theorem

Let $x$ and $y$ be ordinals such that $x < y$.

Then:

$\map V x \in \map V y$

$\map V x \subset \map V y$

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Proof

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The proof shall proceed by Transfinite Induction on $y$.

Basis for the Induction

If $y = 0$, then $x \not < y$.

This proves the basis for the induction.

$\Box$

Induction Step

Let $x < y \implies \map V x \in \map V y$.

Then:

\(\ds x\)

\(<\)

\(\ds y^+\)

\(\ds \leadsto \ \ \)

\(\ds x < y\)

\(\lor\)

\(\ds x = y\)

Definition of Successor Set

\(\ds x = y\)

\(\implies\)

\(\ds \map V x \in \powerset {\map V x}\)

Definition of Power Set

\(\ds \leadsto \ \ \)

\(\ds \map V x\)

\(\in\)

\(\ds \map V {y^+}\)

Definition of Von Neumann Hierarchy

\(\ds x < y\)

\(\implies\)

\(\ds \map V x \in \map V y\)

Inductive Hypothesis

\(\ds \leadsto \ \ \)

\(\ds \map V x\)

\(\in\)

\(\ds \map V {y^+}\)

Von Neumann Hierarchy is Supertransitive

In either case:

$\map V x \in \map V {y^+}$

This proves the induction step.

$\Box$

Limit Case

Let $y$ be a limit ordinal.

Let:

$\map V x \in \map V z$

for all $z \in y$ such that $x < z$.

Since $x < y$, it follows that $x < z$ for some $z \in y$ by Limit Ordinal Equals its Union.

By the inductive hypothesis:

$\map V x \in \map V z$

But by Set is Subset of Union of Family:

$\map V z \subseteq \map V y$

Therefore:

$\map V x \in \map V y$

This proves the limit case.

$\Box$

$\map V x \subset \map V y$ follows by Von Neumann Hierarchy is Supertransitive.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.10$