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

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The Feferman–Levy reals remain uncountable

Statement

The real line of the Feferman–Levy model $N$ is uncountable, even though it is the union of the countable sequence $⟨ R_{m} : m < ω ⟩$ of countable sets.

Facts & Assumptions

Given: The Feferman–Levy symmetric model $N$ .

[F1]

The Feferman–Levy reals are a countable union of countable sets supplies the displayed countable-union decomposition.

[F2]

$R$ is uncountable (Cantor's nested intervals, 1874) proves in ZF, without any Choice principle, that the real line admits no surjection from $ω$ .

[F3]

Hereditarily symmetric interpretations form a transitive ZF model states that the symmetric interpretation $N$ is a transitive ZF model.

Proof

technique · direct internal application of Cantor's theorem

1.1

By F3, apply F2 inside the ZF model $N$ . Its canonical real line $R^{N}$ is a complete ordered field there, and the theorem's nested-interval construction uses no Choice. Hence $N ⊨$ “ $R$ is uncountable.”

F2F3

2.1

By F1 the same set $R^{N}$ equals $⋃_{m < ω} R_{m}$ , where the sequence and every layer belong to $N$ and each layer is countable there. Combining that identity with step 1.1 gives the claimed contrast; it does not infer that the union is countable.

F1step 1.1∎

Depends on

Used by

Countable-union and omega-one regularity principles fail Corollary
The first Feferman–Levy collapse layers Example

Dependency tree · two levels

27 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 and discussion, printed pp. 142–144 (standard reference, not scraped)