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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

1.1F1F2F3construct

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 ${cf}^{N} (ω_{1}^{N}) \leq ω$ .

2.1F2step 1.1∎

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.

Depends on

Dependency tree · 2 levels

Sources

Thomas Jech, The Axiom of Choice, discussion after Theorem 10.6 and Problems 2–3, printed pp. 144, 148