# Von Neumann Hierarchy is Supertransitive

## Theorem

Let $V$ denote the Von Neumann Hierarchy.

Let $x$ be an ordinal.

Then $\map V x$ is supertransitive.

## Proof

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

### Basis for the Induction

We have that:

- $V_0 = \O$

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

This proves the basis for the induction.

$\Box$

### Induction Step

Let $\map V x$ be supertransitive.

First, to prove transitivity.

Let $\map V x$ be transitive.

Then:

\(\displaystyle y\) | \(\in\) | \(\displaystyle \map V {x^+}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y\) | \(\subseteq\) | \(\displaystyle \map V x\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \forall z: \, \) | \(\displaystyle z \in y\) | \(\implies\) | \(\displaystyle z \subseteq \map V x\) | Transitivity of $\map V x$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \forall z: \, \) | \(\displaystyle z \in y\) | \(\implies\) | \(\displaystyle z \in \map V {x^+}\) | Definition of Von Neumann Hierarchy | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y\) | \(\subseteq\) | \(\displaystyle \map V {x^+}\) | Definition of Subset |

Next, to prove supertransitivity:

\(\displaystyle y\) | \(\in\) | \(\displaystyle \powerset {\map V x}\) | Hypothesis | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle y\) | \(\subseteq\) | \(\displaystyle \map V x\) | Definition of Power Set | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \powerset y\) | \(\subseteq\) | \(\displaystyle \powerset {\map V x}\) | Power Set Preserved Under Subset | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map V {x^+}\) | Definition of Von Neumann Hierarchy |

This proves the induction step.

$\Box$

### Limit Case

Let $x$ be a limit ordinal.

Furthermore, let $\map V y$ be transitive for all $y \in x$.

Then:

\(\displaystyle z\) | \(\in\) | \(\displaystyle \map V x\) | Hypothesis | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \exists y \in x: \, \) | \(\displaystyle z\) | \(\in\) | \(\displaystyle \map V y\) | Definition of Von Neumann Hierarchy | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \exists y \in x: \, \) | \(\displaystyle z\) | \(\subseteq\) | \(\displaystyle \map V y\) | Transitivity of $V_y$ | ||||||||

\(\displaystyle \) | \(\subseteq\) | \(\displaystyle \map V x\) | Set is Subset of Union |

This proves transitivity.

Now, to prove supertransitivity:

\(\displaystyle z\) | \(\in\) | \(\displaystyle \map V x\) | Hypothesis | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \exists y \in x: \, \) | \(\displaystyle z\) | \(\in\) | \(\displaystyle \map V y\) | Definition of Von Neumann Hierarchy | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \exists y \in x: \, \) | \(\displaystyle z\) | \(\subseteq\) | \(\displaystyle \map V y\) | Transitivity of $V_y$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\, \displaystyle \exists y \in x: \, \) | \(\displaystyle \powerset z\) | \(\subseteq\) | \(\displaystyle \map V {y + 1}\) | Power Set Preserved Under Subset | ||||||||

\(\displaystyle \) | \(\subseteq\) | \(\displaystyle \map V x\) | Set is Subset of Union |

This proves the limit case.

$\blacksquare$

## Sources

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