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All-real-set measurability yields an inaccessible inner model

Statement

If a universe satisfies ZF+DC and every set of reals is Lebesgue measurable, then its constructible universe $L$ satisfies ZFC and contains an inaccessible cardinal — indeed the ambient $ω_{1}$ is inaccessible in $L$ . Consequently the assumed universe has a definable inner model of ZFC with an inaccessible cardinal. No arithmetized consistency implication is asserted by this item.

Facts & Assumptions

Given: A universe $V$ with ZF+DC in which every set of reals is Lebesgue measurable.

[F1]

AC implies DC implies countable choice: DC implies Countable Choice.

[F2]

Boldface Sigma-one-three measurability: boldface $Σ_{3}^{1}$ measurability means that every $Σ_{3}^{1} (x)$ set is Lebesgue measurable for every real $x$ ; universal measurability of all real sets immediately implies it, since every $Σ_{3}^{1} (x)$ set is a set of reals.

[F3]

Sigma-one-three measurability makes omega-one inaccessible in L: under ZF+Countable Choice and boldface $Σ_{3}^{1}$ measurability, the ambient $ω_{1}$ is inaccessible in $L$ .

[F4]

Semantic and formal inner-model theorem for L with The constructible universe satisfies AC: for each fixed ZFC axiom, ZF proves that axiom relativized to its constructible class $L$ ; in particular the ambient ZF universe proves internally that $L$ satisfies ZFC. The separate external set-model clause of the first supplier is not applied to the proper class $V$ .

[F5]

Inaccessible and Mahlo cardinals: the definition of inaccessibility, so that the same ordinal certified in [F3] is verified to be uncountable, regular and a strong limit inside $L$ .

Proof

1.1

DC implies Countable Choice by [F1], so the choice hypothesis of [F3] holds in $V$ .

1.2

Every set of reals is measurable, hence every $Σ_{3}^{1} (x)$ set is measurable for every real $x$ by [F2]; thus $V$ satisfies boldface $Σ_{3}^{1}$ measurability.

2.1

By [F3] the ambient $ω_{1}$ is inaccessible in $L$ .

F3step 1.1step 1.2

3.1

Apply the fixed-axiom relativization clause of [F4] inside the given ambient ZF universe. It proves that its definable constructible class $L$ satisfies every ZFC axiom. Step 2.1 already says that the ambient $ω_{1}$ , viewed as an ordinal of $L$ , is inaccessible there; equivalently [F5] verifies inside $L$ that it is uncountable, regular and a strong limit. Hence $L ⊨$ "there is an inaccessible cardinal".

F4F5step 2.1

4.1

The steps above give the semantic conclusion that the ambient $ω_{1}$ is inaccessible in the definable inner model $L ⊨ Z F C$ . The formal-inner-model supplier [F4] expressly supplies no arithmetized consistency transfer, so this proof stops at that exact conclusion.

step 2.1step 3.1F4∎

Depends on

Used by

Exact equiconsistency of universal measurability and an inaccessible Theorem

Dependency tree · two levels

35 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

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