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

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

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Countable elementary submodels and their collapses

Statement

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 $A = {a}$ .

Facts & Assumptions

[F1]

Downward Löwenheim–Skolem with parameters: In ZFC let $M$ be an infinite structure for a finite-arity set signature $L$ . If $\max (∣ L ∣, ℵ_{0}) \leq κ \leq ∣ M ∣$ and $A \subseteq M$ has size at most $κ$ , then some elementary substructure $H ≺ M$ contains $A$ and has size exactly $κ$ . Here $∣ L ∣$ counts nonlogical symbols.

[F2]

Collapse of elementary membership submodels: Let $M$ be a set with $(M, \in) ⊨ E x t e n s i o n a l i t y$ and let $X ≺ (M, \in)$ . In ambient ZF, $\in$ restricted to $X$ is well-founded and extensional. It has a unique transitive collapse $π : X \to \overset{ˉ}{X}$ , and the inverse collapse followed by inclusion is an elementary embedding $\overset{ˉ}{X} \to M$ . Countability is preserved by $π$ .

[F3]

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

Proof

Given: Ambient AC, infinite actual membership structure $M ⊨ E x t e n s i o n a l i t y$ and at most countable $A \subseteq M$ .

1.1F1F3given

The language has one binary membership symbol, so $\max (∣ L ∣, ℵ_{0}) = ℵ_{0}$ . As $M$ is infinite, $ℵ_{0} \leq ∣ M ∣$ in ZFC; the parameter set has size at most $ℵ_{0}$ . F1 therefore applies with $κ = ℵ_{0}$ and gives $A \subseteq X ≺ M$ of size exactly $ℵ_{0}$ . The AC premise F3 is used in this supplier to select Skolem witnesses and the size enumerations.

2.1F2step 1.1∎

F2 applies to the actual membership on $X$ and the Extensionality hypothesis on $M$ , giving its transitive collapse. The collapse bijection transports the countable enumeration of $X$ to its image. For a named parameter $a$ , containment of ${a}$ gives $a \in X$ and does not require every member of $a$ to lie in $X$ .

Depends on

Dependency tree · 2 levels

Sources

Geschke, Models of Set Theory — Theorem 4.4 and Corollary 4.6 pp11–12