TheoremStatement: AI-adaptedProof: AI-generatedPipeline-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.

Existence and uniqueness of set satisfaction

Statement

For every set signature $L$ and $L$ -structure $M$ there is a unique set ${Sat}_{M} \subseteq {Form}_{L} \times M^{ω}$ . Writing $M, s ⊨ ϕ$ for membership, its clauses are

$M, s ⊨ u = v ⟺ ⟦ u ⟧_{s} = ⟦ v ⟧_{s},$ $M, s ⊨ R (t_{1}, \dots, t_{n}) ⟺ (⟦ t_{1} ⟧_{s}, \dots, ⟦ t_{n} ⟧_{s}) \in R^{M},$ $M, s ⊨ \neg ϕ ⟺ not M, s ⊨ ϕ,$ $M, s ⊨ ϕ \land ψ ⟺ (M, s ⊨ ϕ and M, s ⊨ ψ),$ $M, s ⊨ \exists x ϕ ⟺ \exists b \in M (M, s [x : = b] ⊨ ϕ) .$

The set is uniformly first-order definable from $L$ and the structure data.

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement.

[F1]

For an $L$ -structure $M$ and assignment $s \in M^{ω}$ , define $⟦ v_{i} ⟧_{s} = s (i)$ , $⟦ c ⟧_{s} = c^{M}$ and $⟦ f (t_{1}, \dots, t_{n}) ⟧_{s} = f^{M} (⟦ t_{1} ⟧_{s}, \dots, ⟦ t_{n} ⟧_{s}) .$ This is a simultaneous definition for all assignments: recurse on terms into the set $M^{(M^{ω})}$ of functions from assignments to $M$ . Variable leaves are projections, constant leaves are constant maps, and each function operation acts pointwise. Structural recursion gives a unique evaluation. The operations are total because the interpreted functions are total. Superscripts $M$ on brackets will distinguish structures when needed. Conventions and prerequisites: def-set-structures-and-variable-assignments, thm-structural-recursion-on-set-coded-syntax. (Term denotation)

Proof

1.1

Put $S = M^{ω}$ . Term denotations give each atomic formula its truth subset of $S$ by Separation: equality compares the two values and a relation tests its interpreted tuple.

2.1

On truth subsets use complement in $S$ , intersection, and for binder $x$ the operation $Q_{x} (B) = {s \in S : \exists b \in M s [x : = b] \in B}$ . These are total operations on the set $P (S)$ . The structural recursion used to define denotation also applies to formulas with atoms as leaves, giving truth sets $B_{ϕ}$ .

F1step 1.1construct

3.1

Separate ${(ϕ, s) : s \in B_{ϕ}}$ inside ${Form}_{L} \times S$ . The displayed clauses follow directly from the operations. Any rival satisfies the same atomic and constructor rules, so structural induction forces equality of all truth sets. All operations and the assertion of existence of the unique recursive evaluation are first-order formulas on sets with parameters $L, M$ ; this yields uniform definability.

step 1.1step 2.1∎

Depends on

Term denotation

Used by

Coincidence for term values and satisfaction Lemma
Skolem witness closure on a cardinal Lemma
Isomorphisms preserve satisfaction Proposition
Relativization agrees with induced set satisfaction Theorem

Dependency tree · two levels

3 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

Moschovakis, Lecture Notes in Logic (2014) — 1C.7–1C.8, pp.11–12; Appendix app4. (standard reference, not scraped)