Dissertations, Theses, and Capstone Projects

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Gunter Fuchs

Committee Members

Arthur Apter

Russell Miller

Subject Categories

Set Theory

Keywords

large cardinals, maximality principle

Abstract

I introduce a new family of axioms extending ZFC set theory, the Sigma_n-correct forcing axioms. These assert roughly that whenever a forcing name a' can be forced by a poset in some forcing class Gamma to have some Sigma_n property phi which is provably preserved by all further forcing in Gamma, then a' reflects to some small name such that there is already in V a filter which interprets that small name so that phi holds. Sigma_1-correct forcing axioms turn out to be equivalent to classical forcing axioms, while Sigma_2-correct forcing axioms for Sigma_2-definable forcing classes are consistent relative to a supercompact cardinal (and in fact hold in the standard model of a classical forcing axiom constructed as an extension of a model with a supercompact), Sigma_3-correct forcing axioms are consistent relative to an extendible cardinal, and more generally Sigma_n-correct forcing axioms are consistent relative to a hierarchy of large cardinals generalizing supercompactness and extendibility whose supremum is the first-order version of Vopenka's Principle.

By analogy to classical forcing axioms, there is also a hierarchy of Sigma_n-correct bounded forcing axioms which are consistent relative to appropriate large cardinals. At the two lowest levels of this hierarchy, outright equiconsistency results are easy to obtain. Beyond these consistency results, I also study when Sigma_n-correct forcing axioms are preserved by forcing, how they relate to previously studied axioms and to each other, and some of their mathematical implications.

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