LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A bounded hierarchy for the multiple-choice argument

Statement

In ZF, for every set $A$ there is a limit ordinal $θ$ such that $A \subseteq V_{θ}$ .

Facts & Assumptions

Given: A set $A$ .

[F1]

The cumulative hierarchy consists of set-sized, transitive, increasing stages (The cumulative hierarchy, Transitivity and growth of hierarchy stages).

[F2]

Foundation implies that every set belongs to some cumulative-hierarchy stage (Equivalent forms of Foundation).

[F3]

$ω$ is a limit ordinal, and for every ordinal $β$ the ordinal $β + ω$ is a limit ordinal with $β \leq β + ω$ ( $ω$ is the least limit ordinal, Ordinal addition $α + β$ , Monotonicity of ordinal $+$ and $\cdot$ : strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities $0 + β = β$ and $1 \cdot β = β$ ).

Proof

technique · direct

1.1

By hierarchy exhaustion, choose an ordinal $β$ with $A \in V_{β}$ .

2.1

The stage $V_{β}$ is transitive, so $A \in V_{β}$ implies $A \subseteq V_{β}$ .

step 1.1F1

3.1

Put $θ = β + ω$ . Then $θ$ is a limit ordinal and $β \leq θ$ ; monotonicity of the hierarchy gives $V_{β} \subseteq V_{θ}$ , hence $A \subseteq V_{θ}$ .

step 2.1F1F3∎

Depends on

Used by

Multiple choice is equivalent to AC in ZF Theorem

Dependency tree · two levels

27 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

Jech, The Axiom of Choice, Theorem 9.1(b), p.134; bounded hierarchy used in the proof (standard reference, not scraped)
Caicedo, Some choiceless results (5), powersets-of-ordinals theorem and hierarchy proof (standard reference, not scraped)