Develops reduction complexities by interpolating effective and Weihrauch reducibility for set-theoretic statements of arbitrary quantifier complexity, with many such complexities independent of ZFC.
The Ground Axiom
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abstract
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.
fields
math.LO 1years
2025 1verdicts
UNVERDICTED 1representative citing papers
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Reduction Complexities in Set Theory
Develops reduction complexities by interpolating effective and Weihrauch reducibility for set-theoretic statements of arbitrary quantifier complexity, with many such complexities independent of ZFC.