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.

Countable transitive models of fixed finite fragments

Statement

For every fixed external finite $Γ \subseteq Z F C$ , ZFC proves that $Γ$ has a countable transitive model. Under ambient AC the same construction applies to fixed finite $Γ \subseteq Z F$ . It does not assert a model of the whole theory or a uniform internal model-existence statement for all coded fragments.

Facts & Assumptions

[F1]

Transitive models of fixed finite axiom fragments: 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.

[F2]

Countable elementary submodels and their collapses: In ZFC, if an infinite set membership structure $M$ satisfies Extensionality, then for every at most countable $A \subseteq M$ there is a countably infinite $X ≺ M$ containing $A$ , and $X$ has a countable transitive collapse. To retain a set $a \in M$ as one parameter, use $A = {a}$ .

[F3]

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

Proof

Given: A fixed external finite fragment and ambient ZFC.

1.1F1F3given

Enlarge the fixed finite fragment by Extensionality, and use F1 to reflect it to $V_{β}$ for $β > ω$ . This is an infinite transitive membership structure satisfying Extensionality and every original axiom of $Γ$ . In the ZFC branch, ambient AC (F3) supplies a reflected Choice axiom if present.

2.1F2step 1.1∎

Apply F2 with empty parameter set to obtain a countable elementary submodel of this stage and its transitive collapse $C$ . Elementarity preserves each sentence of $Γ$ , and the collapse isomorphism preserves the same sentences. Thus $(C, \in) ⊨ Γ$ . AC is used in F2 even when $Γ \subseteq Z F$ ; the earlier reflection of ZF axioms alone does not use it.

Depends on

Dependency tree · 2 levels

Sources

Geschke, Models of Set Theory — Corollary 4.6 pp11–12