TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 2 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Under choice, every metric space has a σ-discrete basis and ‡ Under choice, a regular T₁ space with a σ-locally-finite basis has a compatible normal sequence. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The consistency-strength sandwich for NMSC

Statement

$Con (Z F C + N M S C)$ implies $Con (Z F C + there is a measurable cardinal)$ , while $Con (Z F C + there is a strongly compact cardinal)$ implies $Con (Z F C + N M S C)$ (A strongly compact cardinal gives the NMSC consistency upper bound, Metatheoretic consistency lower bound for NMSC).

Facts & Assumptions

Given: The two metatheoretic consistency hypotheses.

[F1]

The lower bound: $Con (Z F C + N M S C)$ implies $Con (Z F C + a measurable cardinal)$ (Metatheoretic consistency lower bound for NMSC).

[F2]

The upper bound: $Con (Z F C + a strongly compact cardinal)$ implies $Con (Z F C + N M S C)$ (A strongly compact cardinal gives the NMSC consistency upper bound).

[F3]

The internal assertion " $Z F C + N M S C$ proves that there is an inner model with a measurable cardinal" (NMSC gives an inner model with a measurable cardinal) is a distinct claim from both consistency implications. [given]

Proof

technique · direct

1.1

Assume $Con (Z F C + N M S C)$ : by [F1], $Con (Z F C + a measurable cardinal)$ follows.

givenF1

1.2

Assume $Con (Z F C + a strongly compact cardinal)$ : by [F2], $Con (Z F C + N M S C)$ follows.

givenF2

2.1

Steps 1.1 and 1.2 are the two implications of the Statement, and [F3] keeps the internal measurable-inner-model consequence separate from them.

step 1.1step 1.2F3∎

Remarks

Three distinct claims. The internal theorem, the lower consistency implication and the forcing/measure upper implication are kept as three separate assertions; no converse of the upper bound is claimed here (A strongly compact cardinal gives the NMSC consistency upper bound).

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

Fremlin, Real-valued-measurable cardinals (standard reference, not scraped)
Bagaria and da Silva, omega-one-strongly compact cardinals and normality (standard reference, not scraped)