CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Relative consistency of the Feferman–Levy choice failures over ZF

Statement

If ZF is consistent, then so is ZF together with the assertions that the reals are a countable union of countable sets, $cf (ω_{1}) = ω$ , and $\neg {A C}_{ω}$ .

Facts & Assumptions

Given: The fixed arithmetizations of the displayed first-order theories and the hypothesis $Con (Z F)$ .

[F1]

The Feferman–Levy collapse argument is finitely formalizable proves, for every externally fixed finite target fragment, that ZFC+GCH proves the existence of a set model of that fragment.

[F2]

Formal consistency of ZFC plus GCH relative to ZF proves $Con (Z F) \to Con (Z F C + G C H)$ without assuming a transitive set model of ZF.

Proof

technique · contradiction by the finite support of formal derivations

1.1

By F2, the given hypothesis implies $Con (Z F C + G C H)$ . Suppose for contradiction that the target theory $T$ in the Statement is inconsistent. A formal refutation is a finite sequence, so it uses only a finite set $Δ$ of ZF axiom instances together with the three extra target sentences.

F2assume-contra

2.1

Apply F1 to exactly this externally fixed $Δ$ . ZFC+GCH proves that there is a set structure satisfying every sentence used by the alleged refutation. The first-order soundness proof for that finite derivation then proves in ZFC+GCH that the structure satisfies a contradiction, while equality logic proves that no structure does. Hence ZFC+GCH would be inconsistent, contrary to step 1.1.

F1step 1.1discharge-contradiction

3.1

Therefore $T$ is consistent. The argument uses the finite set of formulas occurring in one hypothetical proof; it does not construct a set model of full ZF, invoke semantic completeness, or infer consistency from a merely external citation.

step 1.1step 2.1discharge-contradiction: step 1.1∎

Depends on

Used by

ZF proves that countable unions of countable sets are countable False statement
ZF proves that omega one is regular False statement

Dependency tree · two levels

10 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

Thomas Jech, The Axiom of Choice, Theorem 10.6, printed pp. 142–144, and Problems 2–3, p. 148 (standard reference, not scraped)