False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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False: ZFC proves the normal Moore space conjecture

Statement

Relative to $Con (Z F C)$ , it is false that $Z F C$ proves the normal Moore space conjecture: $Z F C$ does not prove that every normal Moore space is metrizable.

Facts & Assumptions

Given: The metatheoretic hypothesis $Con (Z F C)$ and the fixed arithmetization of The standard certified provability predicate.

[F1]

$Con (Z F C)$ implies $Con (Z F C + C H)$ (Positive relative consistency of CH and GCH).

[F2]

Weakening, concatenation, and replacement of proved sentence premises by their proofs preserve derivability (Finite support, weakening, and composition of derivations).

[F3]

$Z F C + C H$ proves that there is a normal nonmetrizable Moore space (CH yields a normal nonmetrizable Moore space, Moore spaces and developments).

[F4]

$N M S C$ is the assertion that every normal Moore space is metrizable. [given]

Refutation

technique · direct

1.1

Assume $Con (Z F C)$ . Then $Con (Z F C + C H)$ directly by [F1].

givenF1

2.1

If $Z F C$ proved NMSC, weakening would give the same theorem in $Z F C + C H$ . But [F3] gives in that theory a normal nonmetrizable Moore space, contradicting NMSC; by the proof-composition operations of [F2], these two finite derivations concatenate to a $Z F C + C H$ refutation, contrary to [step 1.1].

step 1.1F2F3F4

3.1

Therefore, assuming $Con (Z F C)$ , no such refutation exists and $Z F C$ does not prove NMSC.

step 2.1discharge-contradiction∎

Remarks

The consistency hypothesis cannot be dropped. The conclusion is a relative statement; $Z F C$ itself cannot prove $Con (Z F C)$ (The standard certified provability predicate).
The failure is not a theorem of $Z F C$ alone. The witness space is produced under CH; under PMEA every normal Moore space is metrizable (PMEA implies the normal Moore space conjecture), so the statement is independent in the usual relative-consistency sense.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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)