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The ground axiom

Published online by Cambridge University Press:  12 March 2014

Jonas Reitz
Jonas Reitz*
Affiliation:
The New York City College of Technology, Mathematics, 300 Jay Street, Brooklyn, NY 11201, USA. E-mail: jonasreitz@gmail.com
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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.

Keywords

Forcingcodingordinal definabilitythe Ground Axiomthe Bedrock Axiom
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1299 - 1317
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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