5 research outputs found
Some Intuition behind Large Cardinal Axioms, Their Characterization, and Related Results
We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker
Square compactness and the filter extension property
We show that the consistency strength of κ being 2κ-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juhasz
Set Theory with Urelements
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.Comment: arXiv admin note: text overlap with arXiv:2212.13627. Definition 15
in the previous versions is flawed, which is fixed in this versio
Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
After reviewing various natural bi-interpretations in urelement set theory,
including second-order set theories with urelements, we explore the strength of
second-order reflection in these contexts. Ultimately, we prove, second-order
reflection with the abundant atom axiom is bi-interpretable and hence also
equiconsistent with the existence of a supercompact cardinal. The proof relies
on a reflection characterization of supercompactness, namely, a cardinal
is supercompact if and only if every sentence true in a
structure (of any size) containing in a language of size less than
is also true in a substructure of size less than
with .Comment: 35 pages, 6 figures. Commentary can be made on the first author's
blog at
http://jdh.hamkins.org/second-order-reflection-with-abundant-urelement