The Ground Axiom (GA)
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.
Keywords
Cite
@article{arxiv.math/0609270,
title = {The Ground Axiom (GA)},
author = {Jonas Reitz},
journal= {arXiv preprint arXiv:math/0609270},
year = {2007}
}
Comments
25 pages; v2: corrected typos, removed some standard arguments from proofs