Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13
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.

Semantic generic extensions of countable transitive models

Statement

In ambient ZF, suppose M is an externally countable transitive model of ZFC, P is a nonempty forcing preorder in M, and p∈P. There exists an M-generic G containing p. The resulting M[G] is externally countable, transitive, satisfies ZFC, has the same ordinals as M, and satisfies a fixed formula at ground names exactly when some member of G forces it over M. This theorem is conditional on the full CTM; it does not assert that one exists.

Facts & Assumptions

Given: The stated CTM M, its preorder P, and a condition p. Ground-model AC is part of M satisfying ZFC, not an ambient assumption.

[F1]

Generics over countable transitive models in ZF constructs a generic through any p from a supplied external enumeration of M in ZF.

[F2]

Forcing theorem supplies definability and the truth lemma.

[F3]

Generic extensions satisfy ZF and preserve ground-model Choice gives transitive ZF extensions and propagates ground-model AC.

[F4]

Forcing preserves ordinals gives equality of ordinal heights.

[F5]

Transitive models and finite-fragment transfer data defines external countability by an injection into omega and separates full CTMs from finite fragments.

[F6]

The Axiom of Choice is assumed inside M and used only through the AC branch of F3.

Proof

1.1

From external countability choose one injection j:M→ω. Since M contains the empty set, define e(n) to be the unique x∈M with j(x)=n if there is one, and empty otherwise. This is a surjection e:ω→M. F1 constructs G through p by least enumeration indices, so ambient Choice is unnecessary.

F1F5
2.1

Apply F3 to M and this G. It gives transitivity and ZF, and the fact that M satisfies F6 licenses exactly its ground-name well-ordering step to obtain AC in M[G]. F4 then gives the same ordinals, and F2 gives the asserted equivalence between truth and a condition of G forcing the formula.

F2F3F4F6step 1.1
2.2

Define h(n)=e(n)G when e(n) is a P-name and h(n)=∅ otherwise. External Separation and Replacement make h a function on omega. Every member of M[G] is a value of a name in M, hence appears in h. Assigning each member its least preimage under h injects M[G] into omega. This proves external countability even when distinct names have the same value.

F3step 1.1
3.1

Steps 1.1–2.2 prove all conclusions from the supplied CTM. The single injection j was part of the external countability hypothesis; no collection of CTMs or full-theory model was constructed from a consistency assertion. The only use of AC was the internal one in step 2.1.

F5step 1.1step 2.1step 2.2∎

Depends on

Used by

Dependency tree · two levels

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Sources