Transitive Set Contained in Von Neumann Hierarchy Level
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Theorem
Let $G$ be a transitive set.
Then for some ordinal $i$, $G \subseteq V_i$.
Proof
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Aiming for a contradiction, suppose for each ordinal $i$ the set $G \setminus V_i$ is non-empty.
Let $i$ be any ordinal.
Then by the axiom of foundation:
$\exists x: x \in G\setminus V_i \text{ and } x \cap \paren {G \setminus V_i} = \O$
Since $G$ is transitive, $x \subseteq G$.
Since $x \subseteq G$ and $x \cap \paren {G \setminus V_i} = \O$:
- $x \subseteq V_i$
By the definition of $V$, $x \in V_{i + 1}$, so:
- $x \notin G \setminus V_{i + 1}$
Thus $G \setminus V_{i + 1} \subsetneqq G \setminus V_i$ for each ordinal $i$.
Let $H: \On \to \powerset G$ be defined by:
- $\map H i = G \setminus V_i$ for each ordinal $i$
Since the class of ordinals is well-ordered, applying Strictly Increasing Mapping on Well-Ordered Class proves that for any ordinals $p$ and $q$,
- $p < q$ implies $\map H q \subsetneq \map H p$.
Since the class of ordinals is totally ordered, for any two distinct ordinals $i$ and $j$:
- $\map H i \ne \map H j$
so $H$ is injective.
But that contradicts the fact that $\powerset G$ is a set.
$\blacksquare$