Journal of Symbolic Logic

The ground axiom

Jonas Reitz

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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.

Article information

Source
J. Symbolic Logic, Volume 72, Issue 4 (2007), 1299-1317.

Dates
First available in Project Euclid: 18 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1203350787

Digital Object Identifier
doi:10.2178/jsl/1203350787

Mathematical Reviews number (MathSciNet)
MR2371206

Zentralblatt MATH identifier
1135.03018

Subjects
Primary: 03E35: Consistency and independence results

Keywords
Forcing coding ordinal definability the Ground Axiom the Bedrock Axiom

Citation

Reitz, Jonas. The ground axiom. J. Symbolic Logic 72 (2007), no. 4, 1299--1317. doi:10.2178/jsl/1203350787. https://projecteuclid.org/euclid.jsl/1203350787


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