The ground axiom

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.

• Type: Journal
• Year: 2007
• Where: JSYML
• Authors: Jonas Reitz
• Keywords: Generalized Continuum Hypothesis, Ground Axiom, JSYML 2007, Well-known Set-theoretic Assertions