CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Transitive models of fixed finite axiom fragments

Statement

For each fixed external finite $Γ \subseteq Z F$ , ZF proves that some transitive $V_{β}$ satisfies $Γ$ , with $β$ above any prescribed ordinal bound. In ZFC the analogous scheme holds for fixed finite $Γ \subseteq Z F C$ . These are schemes indexed by external fragments, not a single internal assertion of models for all coded fragments.

Facts & Assumptions

[F1]

Montague–Lévy reflection for a finite formula family: In ZF, for each fixed finite family $Φ$ and every ordinal $α$ , some $β > α$ makes $Φ$ absolute between $V_{β}$ and $V$ , for all tuples in $V_{β}$ . More generally the same holds between $W_{β}$ and $W$ for a definable increasing continuous hierarchy of sets exhausting a definable nonempty class $W$ . For an empty class, the relativization statement is interpreted as a scheme rather than satisfaction in an empty structure.

[F2]

The set of first-order ZF axiom sentences: Let $T_{Z F}$ contain the codes of exactly the following six sentences, together with all instances of the two schemas below.

[F3]

The Axiom of Choice: Every family of nonempty sets has a choice function

Proof

Given: A fixed external finite fragment of ZF, or of ZFC with ambient AC, and an ordinal bound.

1.1F2F3given

List the finitely many sentences $γ_{1}, \dots, γ_{k}$ of $Γ$ using the axiom serialization in F2. Each is an axiom of the ambient theory, so their finite conjunction is a theorem there. If the ambient theory is ZFC and one of these sentences is Choice, use F3 exactly for that sentence. No Choice premise is needed for the ZF branch.

2.1F1step 1.1∎

Apply F1 to that fixed finite list and the desired bound. It gives a $β$ such that each $γ_{i}^{V_{β}} \leftrightarrow γ_{i}$ . Step 1.1 gives the right-hand sides, hence every relativized axiom. The cumulative stage is transitive, so it is the required transitive model. For an infinite carrier begin with a bound at least $ω$ . For $k = 0$ any nonempty stage above the bound works.

Depends on

Dependency tree · 2 levels

Sources

Geschke, Models of Set Theory — Theorem 4.3 application p11