TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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The basic Cohen model has an infinite Dedekind-finite set of reals

Statement

$A$ is infinite and Dedekind-finite. For $A$ , no countably infinite subset, no injection from $ω$ , and no bijection with a proper subset are equivalent and all hold.

Facts & Assumptions

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

[F1]

The Cohen reals form a symmetric set but their enumeration is not symmetric gives $a_{n} \in A$ , pairwise distinctness, and infinitude of $A$ .

[F2]

The basic Cohen set has no countably infinite subset gives the first two negative properties.

[F3]

Dedekind-infinite and Dedekind-finite sets defines a bijection with a proper subset.

[F4]

Dedekind infinitude is equivalent to a countable subset proves the equivalences in ZF.

Proof

1.1

For every $k$ , the finite set ${a_{0}, \dots, a_{k - 1}}$ belongs to the model and has $k$ distinct members, so $A$ is not finite. This uses each finite initial collection, not the absent full enumeration.

2.1

F2 says there is no injection from $ω$ and no countably infinite subset. F4 identifies either positive condition with Dedekind infinitude as defined by F3. Negating the equivalent clauses shows that there is no bijection from $A$ to a proper subset and that $A$ is Dedekind-finite. No Choice is used.

F2F3F4∎

Depends on

Used by

Dependency tree · two levels

8 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, discussion after Theorem 10.25 (standard reference, not scraped)