The Ground Axiom (GA)

Details The Ground Axiom (GA) The Ground Axiom (GA)

2006-09-10

arxiv.org/abs/math/0609270v2

Mathematics

Logic

25 pages; v2: corrected typos, removed some standard arguments from proofs

Scientific paper

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.

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