TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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H_kappa is a transitive subset of V_kappa

Statement

In ZF, for every infinite initial ordinal $κ$ , $H_{κ}$ is a transitive set and $H_{κ} \subseteq V_{κ}$ . For infinite initial ordinals $κ \leq μ$ , one has $H_{κ} \subseteq H_{μ}$ .

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

In ZF, if a transitive set $T$ injects into an ordinal $λ < κ$ , where $κ$ is an infinite initial ordinal, then $rank (t) < κ$ for every $t \in T$ . No regularity or Choice is assumed. (A small transitive set bounds its ranks)

[F2]

In ZF, for every set $x$ and ordinal $α$ , $x \in V_{α} ⟺ rank (x) < α, x \subseteq V_{α} ⟺ rank (x) \leq α .$ Thus $rank (x)$ is the least $α$ with $x \subseteq V_{α}$ , and $rank (x) = α$ iff $x \in V_{α + 1} ∖ V_{α}$ . (Rank characterizes hierarchy membership)

Proof

1.1F1F2

For $x \in H_{κ}$ , let $T = TC ({x})$ and use its witnessing injection into $λ < κ$ . This is a transitive set containing $x$ as an element, so the small-ranks lemma gives $rank (x) < κ$ . Rank characterization puts $x \in V_{κ}$ . Separation inside $V_{κ}$ now proves that $H_{κ}$ is a set.

2.1F1step 1.1

If $y \in x \in H_{κ}$ , the transitive set $TC ({x})$ contains $y$ , and hence contains $TC ({y})$ by minimality. Restrict the same witnessing injection to this smaller closure. Thus $y \in H_{κ}$ , proving transitivity.

3.1F1∎

If $κ \leq μ$ , any witnessing ordinal $λ < κ$ also satisfies $λ < μ$ . The same injection proves $x \in H_{μ}$ .

Depends on

Used by

Hereditary size exhausts V under Choice Corollary
Hₒmega equals Vₒmega Proposition

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weiss, An Introduction to Set Theory (2014) — Theorem 41(1), Exercise 34(1)–(3), p.101. (standard reference, not scraped)