Here is a chart showing the consistency strength and implicational relationships among the large cardinals between supercompact and almost-huge. This is a summary of many of the ideas from chapter two of my dissertation, and it is adapted from a chart appearing in that chapter.
UPDATED June 2014 to reflect my work with Lubarsky on extendible, hypercompact, and enhanced supercompact cardinals.
This is my doctoral dissertation, submitted to the CUNY Graduate Center in April 2013. I plan to produce two journal papers summarizing and extending the material in the dissertation, one paper for each chapter. You can read it here.
Advisor: Joel David Hamkins
Abstract
This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.
The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given either by the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved by forcing in both directions under fairly general assumptions but not in all cases. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.
The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\{j(f)(\kappa) : \ \ \ f: \kappa \to \kappa\}$. Two of the most important results in the chapter are as follows. A Vopěnka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.
Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter
Annals of Pure and Applied Logic, Volume 163, Issue 12, December 2012, Pages 1872-1890.
Abstract
We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.
As of April 23, 2013, I have completed all requirements for the Ph.D.
Huzzah!
The new header photo shows me with my committee immediately after my defense. The defense was on April 12; after that, I had to make final corrections to my dissertation, submit an official hard copy and electronic copy, fill out some surveys and paperwork.
And now . . . I finally have time to work on this Web site, which I haven’t updated since last fall.
Today, I defended my dissertation. You can view the slides from the talk here.
Abstract
This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.
The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given by either the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved by forcing in both directions under fairly general assumptions but not in all cases. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.
The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding j: V –> M such that M is closed under sequences of length equal to the clearance of the embedding. This clearance is defined as the supremum, over all functions f from kappa to kappa, of j(f)(kappa). Two of the most important results in the chapter are as follows. A Vopěnka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.
(This post has been backdated.)
So, I just spent the last hour trying to figure out why I couldn’t get Synctex to work. Synctex is the part of LaTeX that allows one to do inverse search, among other things. In other words, it lets me click on text in the output pdf file and jump immediately to the corresponding LaTeX code. I had two LaTeX files for which synctex wasn’t working. When I compiled other files using the same settings, it worked fine. Finally, I figured out the problem — if the file name contains more than one word separated by spaces, synctex doesn’t work (at least in TexStudio). Well, at least I figured it out. But I wish I could have been spending that time working on my research statement instead.
This is part of why I didn’t major in computer science. I feel like I would have spent half my time searching for bugs like this. Then again, sometimes I wonder whether the problem is worse in math. Math proofs likely contain just as many bugs as computer programs, but since they are checked by fallible humans, some of these bugs are never discovered. But I take comfort in thinking that most such bugs are not major mathematical errors, but rather easily correctable oversights.
Set Theory Seminar
CUNY Graduate Center
Room 6417
Friday, September 7
10 A.M. – 11:45 A.M.
I present a tentative result that Woodinized supercompact cardinals (also known as Woodin for supercompactness cardinals) are equivalent to Vopenka cardinals. This result is vaguely hinted at, though not proven, in Kanamori’s text, and I believe I have worked out the details. A cardinal $\kappa$ is Vopenka iff for every collection of kappa many model-theoretic structures with domains elements of $V_\kappa$, there exists an elementary embedding between two of them. A cardinal $\kappa$ is Woodinized supercompact if it meets the definition of a Woodin cardinal, with strongness replaced by supercompactness. That is to say, for every function $f:\kappa \to \kappa$, there exists a closure point $\delta$ of $f$ and an elementary embedding $j:V \to M$ such that $j(\delta)<\kappa$ and $M$ is closed in $V$ under $j(f)(\delta)$ sequences.
Welcome to my new professional Web page on Booles’ Rings. I will be using this page to post information and reflections about my mathematical research and teaching.
I presented many talks prior to the time that I started this Web site, including not only talks at the CUNY Graduate Center but also talks at the international Young Set Theory conference and an invited talk at Colby College. For a listing of these talks, see my CV.
(This post has been backdated so that it appears in the proper location on the page.)