Set Theory with Urelements

Dissertation, University of Notre Dame (2023)
  Copy   BIBTEX

Abstract

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 91,271

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Reflection Principles and Second-Order Choice Principles with Urelements.Bokai Yao - 2022 - Annals of Pure and Applied Logic 173 (4):103073.
The Usual Model Construction for NFU Preserves Information.M. Randall Holmes - 2012 - Notre Dame Journal of Formal Logic 53 (4):571-580.
Set mapping reflection.Justin Tatch Moore - 2005 - Journal of Mathematical Logic 5 (1):87-97.
The 3-Stratifiable Theorems of.Marcel Crabbé - 1999 - Notre Dame Journal of Formal Logic 40 (2):174-182.
The 3-Stratifiable Theorems of $\mathit{NFU} \infty$.Marcel Crabbé - 1999 - Notre Dame Journal of Formal Logic 40 (2):174-182.
On generic extensions without the axiom of choice.G. P. Monro - 1983 - Journal of Symbolic Logic 48 (1):39-52.
Ultrapowers without the axiom of choice.Mitchell Spector - 1988 - Journal of Symbolic Logic 53 (4):1208-1219.
High-Order Metaphysics as High-Order Abstractions and Choice in Set Theory.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (21):1-3.

Analytics

Added to PP
2023-04-08

Downloads
67 (#235,823)

6 months
24 (#111,774)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Bokai Yao
Peking University

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references