Von Neumann Hierarchy is Supertransitive

Theorem

Let $V$ denote the Von Neumann Hierarchy.

Let $x$ be an ordinal.

Then $V \left({x}\right)$ is supertransitive.

Proof

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

Basis for the Induction

We have that:

$V_0 = \varnothing$

and $\varnothing$ is supertransitive by the very fact that it has no elements.

This proves the basis for the induction.

$\Box$

Induction Step

Let $V \left({x}\right)$ be supertransitive.

First, to prove transitivity.

Let $V \left({x}\right)$ is transitive.

Then:

 $\displaystyle y$ $\in$ $\displaystyle V \left({x^+}\right)$ $\displaystyle \implies \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle V \left({x}\right)$ $\displaystyle \implies \ \$ $\, \displaystyle \forall z: \,$ $\displaystyle z \in y$ $\implies$ $\displaystyle z \subseteq V \left({x}\right)$ Transitivity of $V \left({x}\right)$ $\displaystyle \implies \ \$ $\, \displaystyle \forall z: \,$ $\displaystyle z \in y$ $\implies$ $\displaystyle z \in V \left({x^+}\right)$ Definition of Von Neumann Hierarchy $\displaystyle \implies \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle V \left({x^+}\right)$ Definition of Subset

Next, to prove supertransitivity:

 $\displaystyle y$ $\in$ $\displaystyle \mathcal P \left({ V \left({x}\right) }\right)$ Hypothesis $\displaystyle \implies \ \$ $\displaystyle y$ $\subseteq$ $\displaystyle V \left({x}\right)$ Definition of Power Set $\displaystyle \implies \ \$ $\displaystyle \mathcal P \left({y}\right)$ $\subseteq$ $\displaystyle \mathcal P \left({V \left({x}\right)}\right)$ Power Set Preserved Under Subset $\displaystyle$ $=$ $\displaystyle V \left({x^+}\right)$ Definition of Von Neumann Hierarchy

This proves the induction step.

$\Box$

Limit Case

Let $x$ be a limit ordinal.

Furthermore, let $V \left({y}\right)$ be transitive for all $y \in x$.

Then:

 $\displaystyle z$ $\in$ $\displaystyle V \left({x}\right)$ Hypothesis $\displaystyle \implies \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\in$ $\displaystyle V \left({y}\right)$ Definition of Von Neumann Hierarchy $\displaystyle \implies \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\subseteq$ $\displaystyle V \left({y}\right)$ Transitivity of $V_y$ $\displaystyle$ $\subseteq$ $\displaystyle V \left({x}\right)$ Set is Subset of Union

This proves transitivity.

Now, to prove supertransitivity:

 $\displaystyle z$ $\in$ $\displaystyle V \left({x}\right)$ Hypothesis $\displaystyle \implies \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\in$ $\displaystyle V \left({y}\right)$ Definition of Von Neumann Hierarchy $\displaystyle \implies \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle z$ $\subseteq$ $\displaystyle V \left({y}\right)$ Transitivity of $V_y$ $\displaystyle \implies \ \$ $\, \displaystyle \exists y \in x: \,$ $\displaystyle \mathcal P \left({ z }\right)$ $\subseteq$ $\displaystyle V \left({y + 1}\right)$ Power Set Preserved Under Subset $\displaystyle$ $\subseteq$ $\displaystyle V \left({x}\right)$ Set is Subset of Union

This proves the limit case.

$\blacksquare$