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

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The second Fraenkel model has countable pairs without a choice function

Statement

Let $M$ be a transitive ZFA model with a countably infinite atom set $A$ partitioned into pairs $P_{n}$ for $n \in ω$ , and let $G$ be the full group of permutations of $A$ which preserves each $P_{n}$ setwise. In particular, $G$ contains the permutation that swaps the two atoms of any one $P_{n}$ and fixes every other atom. With the normal filter generated by pointwise stabilizers of finite atom sets, the sequence $(P_{n})$ exists in the permutation model, but its range has no choice function; ${A C}_{ω, 2}$ fails.

Facts & Assumptions

Given: The transitive ZFA model, partition, full pair-preserving group, and finite-support normal filter in the Statement.

[F1]

Fraenkel–Mostowski permutation-model theorem gives the permutation model.

[F2]

Choice for pairs and countable finite choice defines the failed choice principle.

Proof

1.1

Every permitted permutation maps each $P_{n}$ to itself, so each pair and the graph ${(n, P_{n}) : n \in ω}$ have empty support and belong to the model. The pairs remain two-element and their range is countable there.

2.1

Suppose $c (n) \in P_{n}$ were a choice function with finite atom support $E$ . Choose $n$ with $P_{n} \cap E = \emptyset$ . The permutation swapping the two atoms of $P_{n}$ and fixing all other atoms belongs to $G$ , fixes $E$ , every $P_{k}$ , and $n$ , but moves $c (n)$ . It must both fix $c$ by support and move its value at $n$ , contradiction. Thus the displayed family witnesses failure of F2. No AC is used.

F2∎

Depends on

Used by

Dependency tree · two levels

5 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

Jech, The Axiom of Choice, §4.4, pp. 48–49 (standard reference, not scraped)