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Set Theory / Mathematical LogicResearchAI-Generated

For which successor cardinals of regular cardinals does a strong failure of club guessing imply the existence of a generic extension where the failure is witnessed by a uniformly definable family?

Related: Jensen's Square Principle, Shelah's Club Guessing Theorem, Todorcevic's Strong Negative Partition Relations at Successors

Club guessing principles assert that for certain infinite cardinals, one can find a sequence of clubs indexed along a cardinal such that the sequence anticipates any club at stationarily many places. Strong failures of club guessing at successor cardinals of regular cardinals have been shown to be forceable, but it remains open whether such failures can always be made to arise from a definable, uniformly structured family in a canonical generic extension, or whether the combinatorial complexity of the failure is intrinsically tied to non-definable or highly non-constructive configurations. The question asks for a precise classification of when the structural witness to the failure can be chosen with high descriptive complexity bounds versus when it must be wild.

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