False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Consistency does not justify a transitive ZFC model

Statement

False proposed implication over ZFC: $Con (Z F C) \to TM (Z F C)$ . Assuming externally $Con (Z F C + Con (Z F C))$ , there is a set model of ZFC in which this implication fails.

Facts & Assumptions

[F1]

The transitive-model consistency-strength gap: Let $S = Z F C + Con (Z F C)$ . Assuming externally Con(S), ZFC does not prove $Con (Z F C) \to TM (Z F C)$ . Moreover $S + \neg TM (Z F C)$ is consistent and has a set model. The stronger external premise Con(S) is retained.

Refutation

Given: External Con(ZFC+Con(ZFC)) and the standard arithmetic Con and actual-membership TM formulas.

1.1

Under exactly the stated external premise, F1 supplies a nonempty set model N of $Z F C + Con (Z F C) + \neg TM (Z F C)$ . This N is the countermodel witness and meets the ZFC hypothesis of the proposed assertion.

F1given

2.1

In N the antecedent Con(ZFC) is true by its added axiom, while the consequent TM(ZFC) is false by the other added axiom. Thus the implication is false in N by its Boolean satisfaction rule. The supplied model can be at most countable, but its relation is not asserted to be actual membership and it is not asserted to be transitive. Neither countability nor mere consistency repairs the proposed conclusion.

step 1.1algebra∎

Depends on

The transitive-model consistency-strength gap

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Moschovakis, Lecture Notes in Logic (2014) — 4C.8 p153, background only; exact countermodel supplied by the local strength-gap theorem (standard reference, not scraped)
Avigad, Computability and Incompleteness (2007) — §4.7 second incompleteness, applied by the owned strength-gap theorem (standard reference, not scraped)