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

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Countable-union and omega-one regularity principles fail

Statement

In the Feferman–Levy model $N$ :

the assertion that every countable union of countable sets is countable is false;
$ω_{1}$ is singular; and
the Axiom of Countable Choice ${A C}_{ω}$ fails.

Facts & Assumptions

Given: The Feferman–Levy model $N ⊨ Z F$ .

[F1]

The Feferman–Levy reals are a countable union of countable sets writes the reals as one countable union of countable layers.

[F2]

The Feferman–Levy reals remain uncountable proves that this union is uncountable.

[F3]

The Feferman–Levy omega one has countable cofinality gives ${cf}^{N} (ω_{1}) = ω$ .

[F4]

Countable choice makes omega-one regular proves in ZF that ${A C}_{ω}$ implies $cf (ω_{1}) = ω_{1}$ .

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) fixes the exact choice principle being refuted.

Proof

technique · direct consequences and one contraposition

1.1

F1 supplies a countable family of countable sets whose union is $R^{N}$ , while F2 says that union is uncountable. This single witness refutes the universal countable-union assertion in $N$ .

F1F2

1.2

By F3, ${cf}^{N} (ω_{1}) = ω < ω_{1}^{N}$ . Hence the internal cardinal $ω_{1}$ is not regular and is therefore singular.

2.1

If $N$ satisfied ${A C}_{ω}$ as defined in F5, F4 applied inside $N$ would give ${cf}^{N} (ω_{1}) = ω_{1}^{N}$ , contrary to step 1.2. Therefore $N ⊨ \neg {A C}_{ω}$ . These are deductions in ZF from explicit witnesses; no Choice principle is used in deriving its own failure.

F4F5step 1.2∎

Depends on

Used by

The Feferman–Levy collapse argument is finitely formalizable Lemma

Dependency tree · two levels

18 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 Problems 2–3, printed pp. 142–144, 148 (standard reference, not scraped)