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

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.

Downward Löwenheim–Skolem with parameters

Statement

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.

Facts & Assumptions

Given: The stated hypotheses, including AC.

[F1]

In ZFC the witness hull of a subset of size at most an infinite $κ$ , in a language of size at most $κ$ , is elementary and has size at most $κ$ . (Skolem hulls are small elementary substructures)

[F2]

The union of two sets of size at most infinite $κ$ has size at most $κ$ , by $κ + κ = κ$ . (Absorption: for cardinals $κ, λ$ with $κ$ infinite and $λ \leq κ$ , $κ \oplus λ = κ$ , and $κ \otimes λ = κ$ when $λ \neq 0$ )

[A1]

AC is assumed. (The Axiom of Choice)

Proof

1.1

The inequality $κ \leq ∣ M ∣$ supplies an injection $i : κ \to M$ . Put $B = A \cup i [κ]$ . Then $A \subseteq B \subseteq M$ , and $∣ B ∣ \leq κ$ by F2, while $i$ witnesses $κ \leq ∣ B ∣$ . Thus $∣ B ∣ = κ$ , including when $A$ is empty or already has size $κ$ .

2.1

Apply F1 to $B$ : its size is $κ$ , $κ$ is infinite, and $∣ L ∣ \leq κ$ . Under A1 this yields an elementary hull $H$ containing $B$ with $∣ H ∣ \leq κ$ . Since $i [κ] \subseteq H$ , also $κ \leq ∣ H ∣$ , so $∣ H ∣ = κ$ , and $A \subseteq H$ . When $κ = ∣ M ∣$ , one may take a bijection $i : κ \to M$ in step 1.1, obtaining $B = H = M$ .

F1A1step 1.1∎

Depends on

Used by

Dependency tree · two levels

19 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–D’Mello, Fundamentals of Model Theory, Theorem 5, printed p.20; equality endpoint included. (standard reference, not scraped)