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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

1.1F1F2

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 $N$ .

1.2F3

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

2.1F4F5step 1.2∎

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.

Depends on

Dependency tree · 2 levels

Sources

Thomas Jech, The Axiom of Choice, Theorem 10.6 and Problems 2–3, printed pp. 142–144, 148