Award Date

5-1-2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Douglas Burke

Second Committee Member

Derrick DuBose

Third Committee Member

Satish Bhatnagar

Fourth Committee Member

Pushkin Kachroo

Number of Pages

38

Abstract

In 1971, Kunen proved that the Axiom of Choice imposes a ceiling on the large cardinal hierarchy [7]. Much like the assumption V ≠ L unlocks measurable cardinals and beyond, dropping the Axiom of Choice enables Reinhardt cardinals and stronger cardinals to be explored. Some major notions of large cardinals beyond choice have recently been standardized by Woodin et. al. [2], with questions raised regarding their interconnectedness. Part 1 of this dissertation partially answers two of those questions, while conjecturing, with a partial solution, a much stronger answer which would simplify the existing cardinal charts - that Regular Berkeley Cardinals are Club Berkeley Cardinals. Part 2 articulates some variations of the major choiceless cardinals, illuminating an iterating structure between them.

Keywords

Axiom of Choice; Berkeley Cardinals; Elementary Embeddings; Large Cardinals; Reinhardt Cardinals; Set Theory

Disciplines

Mathematics

File Format

pdf

File Size

4000 KB

Degree Grantor

University of Nevada, Las Vegas

Language

English

Rights

IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

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Mathematics Commons

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