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

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The Feferman–Levy omega one has countable cofinality

Statement

In the Feferman–Levy model $N$ ,

$cf (ω_{1}) = ω .$

Facts & Assumptions

Given: The transitive model $N$ and its ordinal $ω_{1}^{N}$ .

[F1]

The new omega one is the old aleph omega identifies $ω_{1}^{N}$ with $ℵ_{o} m e g a^{V}$ .

[F2]

$cf (α) \leq α$ ; $cf (0) = 0$ and $cf (α + 1) = 1$ ; for a limit ordinal $λ$ the value $cf (λ)$ is an infinite cardinal with $cf (cf (λ)) = cf (λ)$ , so it is regular; and every cofinal subset of $λ$ has cardinality at least $cf (λ)$ , a value that is attained says in ZF that the cofinality of a limit ordinal is an infinite cardinal and is bounded by the size of every exhibited cofinal subset.

[F3]

Hereditarily symmetric interpretations form a transitive ZF model gives $V \subseteq N$ for this symmetric construction.

Proof

technique · direct computation from an explicit cofinal sequence

1.1

The ground sequence $c (n) = ℵ_{n}^{V}$ is a set of $V$ and hence, by F3, a set of $N$ . Its range is cofinal in $ℵ_{ω}^{V}$ by the definition of the limit aleph. Using F1, $c : ω \to ω_{1}^{N}$ is therefore cofinal in $N$ , so ${cf}^{N} (ω_{1}^{N}) \leq ω$ .

F1F2F3construct

2.1

The ordinal $ω_{1}^{N}$ is an infinite cardinal and therefore a limit ordinal. F2 makes its cofinality an infinite cardinal, hence at least $ω$ . Combined with step 1.1, this gives ${cf}^{N} (ω_{1}^{N}) = ω$ . The witness is the one ground sequence $c$ ; no sequence of arbitrary choices is used.

F2step 1.1∎

Depends on

Used by

Countable-union and omega-one regularity principles fail Corollary

Dependency tree · two levels

29 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, discussion after Theorem 10.6 and Problems 2–3, printed pp. 144, 148 (standard reference, not scraped)