CorollaryStatement: AI-adaptedProof: 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.

Generics over countable transitive models in ZF

Statement

In ZF, let $e : ω \to M$ be an external surjective enumeration of a transitive set model M of ZF. If M contains a forcing preorder P and its order, then for each $p \in P$ an M-generic filter through p exists. The existence of M and its enumeration are hypotheses.

Facts & Assumptions

Given: ZF with supplied external enumeration of M. Enumerate all ground dense sets with P as fallback, refine by least e-indices, and verify the resulting filter directly; no AC-dependent theorem is consumed.

[F1]

Dense open sets and generic filters over a model: Density is absolute, and genericity means meeting every ground-model dense set.

[F2]

Transfinite recursion: A unique definable successor rule gives a sequence on omega in ZF.

Proof

1.1

Set $D_{n} = e (n)$ if e(n) is a dense subset of P, and $D_{n} = P$ otherwise. Every ground-model dense subset occurs. For q in P let $h (n, q) = e (k)$ for the least k such that $e (k) \in D_{n}$ and $e (k) \leq q$ . Transitivity gives $P \subseteq M$ , so surjectivity of e and density ensure such k exists. This is a uniquely defined rule, not a choice function obtained by AC. Recursively put $p_{0} = p$ and $p_{n + 1} = h (n, p_{n})$ .

F1F2construct

2.1

Put $G = {q \in P : \exists n p_{n} \leq q}$ . Then p belongs to G; transitivity gives upward closure; and the term with index max(n,m) strengthens any two members witnessed by n,m. Thus G is a filter. Each $p_{n + 1} \in D_{n} \cap G$ , so F1 makes G M-generic. Neither the sequence nor G is asserted to be an element of M.

F1step 1.1∎

Depends on

Used by

A generic filter belongs to its ground model False statement

Dependency tree · two levels

6 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

Karagila Theorem 1.14 and Corollary 1.15 p4; local least-code ZF refinement (standard reference, not scraped)