Von Neumann Hierarchy is Supertransitive

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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.

$\blacksquare$


Sources