The Ground Axiom

Author

Jonas Reitz, The Graduate Center, City University of New YorkFollow

Date of Degree

2006

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Joel David Hamkins

Committee Members

Sergei Artemov

Melvin Fitting

Attila Mate

Subject Categories

Mathematics

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.

Comments

Digital reproduction from the UMI microform.

Recommended Citation

Reitz, Jonas, "The Ground Axiom" (2006). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/5920