TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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NMSC gives an inner model with a measurable cardinal

Statement

$Z F C + N M S C$ proves that there is an inner model with a measurable cardinal, where NMSC is the assertion that every normal Moore space is metrizable and an inner model is a transitive class model of $Z F C$ containing all ordinals (Semantic and formal inner-model theorem for L).

Facts & Assumptions

Given: The hypothesis NMSC over $Z F C$ .

[F1]

If there is no inner model with a measurable cardinal, then $Z F C$ proves HYP with the parameters the construction consumes (No inner measurable implies Fleissner's HYP, Fleissner's HYP covering interface).

[F2]

HYP proves that there is a normal nonmetrizable Moore space (HYP produces a normal nonmetrizable Moore space, Moore spaces and developments).

[F3]

Such a space is a counterexample to NMSC. [given]

Proof

technique · contrapositive

1.1

Argue contrapositively inside $Z F C$ : assume there is no inner model with a measurable cardinal.

contrapositive-reduceassume-hypgiven

2.1

By [F1] the assumption implies HYP with a fixed parameter triple $κ, (κ_{n}), E$ and fixed ladders.

step 1.1F1

3.1

By [F2] applied to those parameters there is a normal nonmetrizable Moore space.

step 2.1F2

4.1

That space violates NMSC by [F3]. Hence the assumption of step 1.1 implies the negation of NMSC; contraposition gives that NMSC implies the existence of an inner model with a measurable cardinal.

step 3.1F3discharge-contrapositive∎

Remarks

The inner model is not constructed by the topological argument. The topological half produces a counterexample to NMSC from HYP; the existence of the inner model is the contrapositive of the covering-theoretic half (No inner measurable implies Fleissner's HYP). No claim is made here that the space or its construction yields a measurable cardinal.

Depends on

Used by

Metatheoretic consistency lower bound for NMSC Theorem
The consistency-strength sandwich for NMSC Theorem

Dependency tree · two levels

32 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

William G. Fleissner, If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal (standard reference, not scraped)