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Sigma-one-three measurability makes omega-one inaccessible in L

Statement

Assume ZF+Countable Choice and boldface $Σ_{3}^{1}$ measurability. Then the ambient $ω_{1}$ is an inaccessible cardinal in $L$ .

Facts & Assumptions

Given: Countable Choice and the hypothesis that every $Σ_{3}^{1} (x)$ set of reals is Lebesgue measurable for every real $x$ .

[F1]

Boldface Sigma-one-three measurability: the pointclass $Σ_{3}^{1} (x)$ , the inclusion of $Σ_{2}^{1} (x)$ in $Σ_{3}^{1} (x)$ by a dummy real quantifier, and the definition of boldface measurability.

[F2]

Failure of inaccessibility in L produces a real with correct omega-one: failure of inaccessibility of the ambient $ω_{1}$ in $L$ yields a real $x$ with $ω_{1}^{L [x]} = ω_{1}$ .

[F3]

A measurable null-code order bounds the constructible null union: the null-code order $A (x)$ is $Σ_{2}^{1} (x)$ , and its measurability makes the union of the constructible null Borel sets null.

[F4]

Uniform null-code measurability makes the Raisonnier filter rapid: under Countable Choice, $ω_{1}^{L [x]} = ω_{1}$ and measurability of $A (x \oplus r)$ for every real $r$ , the filter $F (x)$ is rapid.

[F5]

The Raisonnier family is a Sigma-one-three filter: $F (x)$ is a $Σ_{3}^{1} (x)$ subset of the reals.

[F6]

Rapid filters are not Lebesgue measurable: rapid filters are not Lebesgue measurable.

[F7]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the ambient choice hypothesis.

Proof

1.1

Assume, for contradiction, that the ambient $ω_{1}$ is not an inaccessible cardinal of $L$ , and let $x$ be a real with $ω_{1}^{L [x]} = ω_{1}$ , as supplied by [F2].

assume-contraF2

2.1

For every real $r$ , the null-code order $A (x \oplus r)$ is $Σ_{2}^{1} (x \oplus r)$ by [F3]. By [F1] it is $Σ_{3}^{1} (x \oplus r)$ , so the boldface measurability hypothesis makes $A (x \oplus r)$ measurable. Thus the uniform hypothesis of [F4] holds, not merely its instance at $r = 0$ .

F1F3step 1.1

3.1

By [F4], applied with Countable Choice [F7], $ω_{1}^{L [x]} = ω_{1}$ and the uniform conclusion of step 2.1, the Raisonnier filter $F (x)$ is rapid.

F4F7step 2.1

4.1

By [F5] the filter $F (x)$ is a $Σ_{3}^{1} (x)$ set of reals, and by [F6] it is not Lebesgue measurable. This contradicts the hypothesis that every $Σ_{3}^{1} (x)$ set is measurable.

F5F6step 3.1

5.1

The contradiction in step 4.1 refutes the assumption of step 1.1, so the ambient $ω_{1}$ is inaccessible in $L$ .

discharge-contradictionstep 4.1∎

Depends on

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Sources

Hiromi Ishii, Regularity Properties and Inaccessible Cardinals (standard reference, not scraped)