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

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Each Feferman–Levy real layer is countable

Statement

For every $m < ω$ , the real layer $R_{m}$ is countable in the Feferman–Levy model $N$ .

Facts & Assumptions

Given: One fixed $m < ω$ and the corresponding layer in $N$ .

[F1]

Each real layer has a ground-model cardinal bound supplies in $N$ a specified surjection $e_{m} : ℵ_{m + 1}^{V} ↠ R_{m}$ .

[F2]

Every finite ground aleph is countable in the Feferman–Levy model says that $ℵ_{m + 1}^{V}$ is countable in $N$ .

[F3]

A nonempty set is at most countable iff it is a surjective image of $N$ says that every nonempty countable set is the range of a surjection from $ω$ , without Choice.

Proof

technique · direct composition of the two supplied maps

1.1

The ordinal $ℵ_{m + 1}^{V}$ is nonempty. By F2 and F3, fix in $N$ one surjection $f : ω ↠ ℵ_{m + 1}^{V}$ , and form $g_{m} = e_{m} \circ f$ . Both factors are sets of $N$ , and ordinary ordered-pair Separation produces their composition. For each $x \in R_{m}$ , its $e_{m}$ -preimage is nonempty, so take its least ordinal member $ξ$ ; then the $f$ -preimage of $ξ$ is a nonempty set of naturals and has a least member $k$ . Thus $g_{m} (k) = x$ , so $g_{m} : ω ↠ R_{m}$ . This fixes one witness for one already fixed $m$ ; it does not choose a family indexed by $ω$ .

F1F2F3construct

2.1

The layer is nonempty because it contains the interpretation of the constantly-zero Boolean name. Sending each $x \in R_{m}$ to its least $g_{m}$ -preimage gives an injection into $ω$ , so $R_{m}$ is at most countable. The least-preimage clauses are definable and involve one fixed map; no choice function for the family $(R_{m})_{m < ω}$ is formed.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

17 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, Lemmas 10.8–10.9 and conclusion of Theorem 10.6, printed p. 144 (standard reference, not scraped)