Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)
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.

A generic filter belongs to its ground model

Statement

False claim, conditional on a supplied externally countable transitive ZF model M: every M-generic filter for a forcing notion in M belongs to M.

Facts & Assumptions

Given: ZF conditional on the supplied countable transitive model and enumeration. Cohen splitting gives atomlessness, a generic exists by least-index recursion, and its failure to be ground-model follows both from the dense-complement theorem and the real-union calculation.

[F1]

Atomless generic filters are not in the ground model: An M-generic filter on atomless forcing is not in M.

[F2]

Generics over countable transitive models in ZF: A supplied external enumeration produces an M-generic filter through any condition in ZF.

[F3]

Cohen-name valuation and dense-set meeting: For Cohen forcing the union of a generic filter is a total binary sequence different from every ground-model binary sequence.

Refutation

1.1

In the supplied M use P=2<ω with extension order. Its finite sequences and order are the actual ones by transitivity and actual omega. Below each s the sequences formed by appending 0 and 1 are incompatible, so P is atomless. Apply F2 through the empty condition to obtain an M-generic filter G.

F2construct
2.1

F1 now gives G∉M, refuting the universal claim. Equivalently, F3 makes ⋃G a binary sequence not in M, whereas G in M would put its actual union in M by internal Union and transitivity. The counterexample remains conditional on the supplied model; ZF does not here prove that such a model exists. Singleton forcing still has a ground-model generic filter, so the false claim is not replaced by an unconditional assertion about all forcing orders.

F1F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources