CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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.

The basic Cohen model fails well-orderability and AC

Statement

The infinite Dedekind-finite set $A$ cannot be well-ordered. Hence the basic Cohen symmetric model satisfies ZF plus $\neg A C$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

The basic Cohen model has an infinite Dedekind-finite set of reals gives infinitude and no $ω$ -injection.

[F2]

The well-ordering theorem says AC well-orders every set.

[F3]

The Axiom of Choice identifies the failed axiom.

[F4]

The basic Cohen symmetric system and Hereditarily symmetric interpretations form a transitive ZF model identify the displayed HS interpretation as the transitive ZF model in which $A$ lives.

Proof

1.1

If $A$ had a well-order, recursion selecting the least unused member would either terminate after finitely many steps—making $A$ finite—or define an injection $ω \to A$ . Both contradict F1. Thus $A$ is not well-orderable.

2.1

By F4 the basic Cohen HS interpretation is a ZF model containing $A$ . If it satisfied F3, F2 inside that model would well-order $A$ , contradicting step 1.1. Hence AC fails. This reductio is the exact use of the choice dependency; the symmetric-model construction itself remains choice-free.

F2F3F4∎

Depends on

Used by

Every symmetric submodel satisfies Choice False statement
Fixed finite-fragment verification for the basic Cohen symmetric model Lemma

Dependency tree · two levels

23 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, Forcing & Symmetric Extensions, Theorem 10.25 (standard reference, not scraped)