Jason Schanker

Jason Aaron Schanker earned his Ph.D. under my supervision at the CUNY Graduate Center in June, 2011.  Jason’s dissertation introduces several interesting new large cardinal notions, investigating their interaction with forcing, indestructibility, the Generalized Continuum Hypothesis and other topics.  He defines that a cardinal $\kappa$ is weakly measurable, for example, if any family of $\kappa^+$ many subsets of $\kappa$ can be measured by a $\kappa$-complete filter.  This is equivalent to measurability under the GCH, of course, but the notions are not equivalent in general, although they are equiconsistent.  The weak measurability concept can be viewed as a generalization of weak compactness, and there are myriad equivalent formulations, including elementary embedding characterizations using transitive domains of size $\kappa^+$.  It was known classically that the failure of the GCH at a measurable cardinal has consistency strength strictly greater than a measurable cardinal, but Jason proved that the corresponding fact is not true for the weakly measurable cardinals.  Generalizing this notion, Jason introduced the near supercompactness hierarchy, which refines and extends the usual supercompactness hierarchy in a way that adapts well to many existing forcing arguments.  Jason holds a faculty position at Manhattanville College in Purchase, New York.

Jason Schanker

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Jason Schanker, “Weakly Measurable Cardinals and Partial Near Supercompactness,”  Ph.D. dissertation for the Graduate Center of the City University of New York, June, 2011.

Abstract.  I will introduce a few new large cardinal concepts. A weakly measurable cardinal is a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal $\kappa$ is weakly measurable if for every collection $A$ containing at most $\kappa^+$ many subsets of $\kappa$, there exists a nonprincipal $\kappa$-complete filter on $\kappa$ measuring all sets in $A$. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if $\kappa$ is measurable, then we can make its weak measurability indestructible by the forcing $\text{Add}(\kappa,\eta)$ for all $\eta$ while forcing the GCH to hold below $\kappa$. Nevertheless, I shall prove that weakly measurable v cardinals and measurable cardinals are equiconsistent.

A cardinal κ is nearly $\theta$-supercompact if for every $A\subset\theta$, there exists a transitive $M\models\text{ZFC}^-$ closed under ${<}\kappa$ sequences with $A,\kappa,\theta\in M$, a transitive $N$, and an elementary embedding $j : M \to  N$ with critical point $\kappa$ such that $j(\kappa) > \theta$ and $j”\theta\in N$. This concept strictly refines the $\theta$-supercompactness hierarchy as every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{\theta^{{<}\kappa}}$-supercompact cardinal $\kappa$ is $\theta$-supercompact. Moreover, if $\kappa$ is a $\theta$-supercompact cardinal for some $\theta$ such that $\theta^{{<}\kappa}=\theta$, we can move to a forcing extension preserving all cardinals below $\theta^{++}$ where $\kappa$ remains $\theta$-supercompact but is not nearly $\theta^+$-supercompact. I will also show that if $\kappa$ is nearly $\theta$-supercompact for some $\theta\geq 2^\kappa$ such that $\theta^{{<}\theta}=\theta$, then there exists a forcing extension preserving all cardinals at or above $\kappa$ where $\kappa$ is nearly $\theta$-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result, and I will prove that if $\kappa$ is nearly $\theta$-supercompact for some $\theta\geq\kappa$ such that $\theta^{{<}\theta}=\theta$, then there is a forcing extension where its near $\theta$-supercompactness is preserved and indestructible by any further ${<}\kappa$-directed closed $\theta$-c.c. forcing of size at most $\theta$. Finally, these cardinals have high consistency strength. Specifically, I will show that if $\kappa$ is nearly $\theta$-supercompact for some $\theta\geq\kappa^+$ for which $\theta^{{<}\theta}=\theta$, then AD holds in $L(\mathbb{R})$. In particular, if $\kappa$ is nearly $\kappa^+$-supercompact and $2^\kappa=\kappa^+$, then AD holds in $L(\mathbb{R})$.

Ansten Mørch-Klev

Ansten Mørch-Klev earned his M.Sc. degree under my direction at Universiteit van Amsterdam in July, 2007.   For his thesis, Ansten undertook to investigate the infinite-time analogue of Kleene’s $\mathcal{O}$, the natural extension of Kleene’s concept to the case of infinite time Turing machines.  The result was a satisfying and robust theory, which revealed (as predicted by Philip Welch) the central importance of the eventually writable ordinals in the theory of infinite time computability.  This work eventually appeared as:  Ansten Mørch-Klev, “Infinite time analogues of Kleene’s $\mathcal{O}$,” Archive for Mathematical Logic, 48(7):2009, p. 691-703, DOI:10.1007/s00153-009-0146-2.

Ansten Mørch Klev

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Ansten Mørch-Klev, “Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter”, M.Sc. thesis for Institute of Logic, Language and Computation, Universiteit van Amsterdam, July, 2007.  ILLC publication

Abstract.  We define two successive extensions of Kleene’s $\mathcal{O}$ using infinite time Turing machines. The first extension, $\mathcal{O}^+$, is proved to code a tree of height $\lambda$, the supremum of the writable ordinals, while the second extension, $\mathcal{O}^{++}$, is proved to code a tree of height $\zeta$, the supremum of the eventually writable ordinals. Furthermore, we show that $\mathcal{O}^+$ is computably isomorphic to $h$, the lightface halting problem of infinite time Turing machine computability, and that $\mathcal{O}^{++}$ is computably isomorphic to $s$, the set of programs that eventually writes a real. The last of these results implies, by work of Welch, that $\mathcal{O}^{++}$ is computably isomorphic to the $\Sigma_2$ theory of $L_\zeta$, and, by work of Burgess, that $\mathcal{O}^{++}$ is complete with respect to the class of the arithmetically quasi-inductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of $\Sigma_2(L_\zeta)$, a theory in which $\mathcal{O}^{++}$ plays the role of $\mathcal{O}$, the arithmetically quasi-inductive sets play the role of $\Pi^1_1$, and the eventually writable reals play the role of $\Delta^1_1$.


Victoria Gitman

Victoria Gitman earned her Ph.D. under my supervision at the CUNY Graduate Center in June, 2007.  For her dissertation work, Victoria had chosen a very difficult problem, the 1962 question of Dana Scott to characterize the standard systems of models of Peano Arithmetic, a question in the field of models of arithmetic that had been open for over forty years. Victoria was able to make progress, now published in several papers, by using an inter-disciplinary approach, applying set-theoretic ideas—including a use of the proper forcing axiom PFA—to the problem in the area of models of arithmetic, where such methods hadn’t often yet arisen.  Ultimately, she showed under PFA that every arithmetically closed proper Scott set is the standard system of a model of PA.  This result extends the classical result to a large new family of Scott sets, providing for these sets an affirmative solution to Scott’s problem.  In other dissertation work, Victoria untangled the confusing mass of ideas surrounding various Ramsey-like large cardinal concepts, ultimately separating them into a beautiful hierarchy, a neighborhood of the vast large cardinal hierarchy intensely studied by set theorists.  (Please see the diagram in her dissertation.)  Victoria holds a tenure-track position at the New York City College of Technology of CUNY.

Victoria Gitman

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Victoria Gitman, “Applications of the Proper Forcing Axiom to Models of Peano Arithmetic,”  Ph.D. dissertation for the Graduate Center of the City University of New York, June 2007.

Abstract. In Chapter 1, new results are presented on Scott’s Problem in the subject of models of Peano Arithmetic. Some forty years ago, Dana Scott showed that countable Scott sets are exactly the countable standard systems of models of PA, and two decades later, Knight and Nadel extended his result to Scott sets of size $\omega_1$. Here it is shown that assuming the Proper Forcing Axiom, every arithmetically closed proper Scott set is the standard system of a model of PA. In Chapter 2, new large cardinal axioms, based on Ramsey-like embedding properties, are introduced and placed within the large cardinal hierarchy. These notions generalize the seldom encountered embedding characterization of Ramsey cardinals. I also show how these large cardinals can be used to obtain indestructibility results for Ramsey cardinals.

Thomas Johnstone

Thomas Johnstone earned his Ph.D. under my supervision in June, 2007 at the CUNY Graduate Center.  Tom likes to get thoroughly to the bottom of a problem, and this indeed is what he did in his dissertation work on the forcing-theoretic aspects of unfoldable cardinals.  He seemed to want always to dig deeper, seeking out the unstated general phenomenon behind the results.  His characteristic style of giving a seminar talk—pure mathematical pleasure to attend—is to explain not only why the mathematical fact is true, but also why the proof must be the way that it is.  Thomas holds a tenure-track position at the New York City College of Technology of CUNY.

Thomas A. Johnstone

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Thomas A. Johnstone, “Strongly unfoldable cardinals made indestructible,” Ph.D. dissertation, The Graduate Center of the City University of New York, June 2007.

Abstract. I provide indestructibility results for weakly compact, indescribable and strongly unfoldable cardinals. In order to make these large cardinals indestructible, I assume the existence of a strongly unfoldable cardinal $\kappa$, which is a hypothesis consistent with $V=L$. The main result shows that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${<}\kappa$-closed forcing which does not collapse $\kappa^{+}$. As strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, I obtain indestructibility for these cardinals also, thereby reducing the large cardinal hypothesis of previously known indestructibility results for these cardinals significantly. Finally, I use the developed methods to show the consistency of a weakening of the Proper Forcing Axiom $\rm PFA$ relative to the existence of a strongly unfoldable cardinal.

Jonas Reitz

Jonas Reitz earned his Ph.D under my supervision in June, 2006 at the CUNY Graduate Center.  He was truly a pleasure to supervise. From the earliest days of his dissertation research, he had his own plan for the topic of the work: he wanted to “undo” forcing, to somehow force backwards, from the extension to the ground model. At first I was skeptical, but in time, ideas crystalized around the ground axiom (now with its own Wikipedia entry), formulated using a recent-at-the-time result of Richard Laver.  Along with Laver’s theorem, Jonas’s dissertation was the beginning of the body of work now known as set-theoretic geology.  Jonas holds a tenured position at the New York City College of Technology of CUNY.

Jonas Reitz

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Jonas Reitz, “The ground axiom,” Ph.D. dissertation, CUNY Graduate Center, June, 2006.  ar$\chi$iv

Abstract.  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.

George Leibman

George Joseph Leibman earned his Ph.D. under my supervision in June, 2004 at the CUNY Graduate Center. He was my first Ph.D. student. Being very interested both in forcing and in modal logic, it was natural for him to throw himself into the emerging developments at the common boundary of these topics.  He worked specifically on the natural extensions of the maximality principle where when one considers a fixed definable class $\Gamma$ of forcing notions.  This research engaged with fundamental questions about the connection between the forcing-theoretic properties of the forcing class $\Gamma$ and the modal logic of its forcing validities, and was a precursor of later work, including joint work, on the modal logic of forcing.

George Leibman

George Leibman


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George Leibman, “Consistency Strengths of Modified Maximality Principles,” Ph.D. thesis, CUNY Graduate Center, 2004.  ar$\chi$iv

Abstract. The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension $V^{\mathbb{P}}$, and remains true in any further forcing extension of $V^{\mathbb{P}}$, then it is true in all forcing extensions of $V$.  A modified maximality principle $\text{MP}_\Gamma$ arises when considering forcing with a particular class $\Gamma$ of forcing notions. A parametrized form of such a principle, $\text{MP}_\Gamma(X)$, considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set $X$ which depends on the forcing class $\Gamma$ being considered. A stronger necessary form of such a principle, $\square\text{MP}_\Gamma(X)$, occurs when it continues to be true in all $\Gamma$ forcing extensions.

This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, COHEN, COLL (the forcing notions that collapse ordinals to $\omega$), ${\lt}\kappa$ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered.

More students (on whose thesis committees I’ve served)

I have served as a member of the dissertation or thesis committee for each the following students.

Konstantinos Tsaprounis, Ph.D. 2012, Universitat de Barcelona, Departament de Lògica, Història i Filosofia de la Ciència, Programa de doctorat de Lògica Pura i Aplicada, Facultat de Filosofia. Barcelona Research Group in Set Theory

Director: Joan Bagaria i Pigrau

Dissertation: Large cardinals and resurrection axioms

In his dissertation, Kostas develops the theory of $C^{(n)}$-tall cardinals, $C^{(n)}$-superstrong, $C^{(n)}$-strong, $C^{(n)}$-strongly compact, $C^{(n)}$-Woodin, $C^{(n)}$-supercompact and $C^{(n)}$-extendible cardinals, particularly with a view to finding upper bounds in consistency strength via an elementary chain construction.  In addition, he investigates various resurrection axioms, including RA(stationary-preserving).

Shoshana Friedman

Shoshana Friedman, Ph.D. 2010, CUNY Graduate Center, math genealogy | MathSciNet

Supervisor: Arthur W. Apter

Dissertation:   Aspects of supercompactness, HOD and set-theoretic geology

Abstract. In this thesis, we study HOD, primarily in the context of large cardinals and GCH. Chapter 1 contains our introductory comments and preliminary remarks. In Chapter 2, we extend a property of HOD-supercompactness due to Sargsyan to various models of set theory containing supercompact cardinals. In doing so, we develop a new method for coding sets while preserving GCH. In Chapter 3, we extend this alternative method of coding. This allows us to produce models of V = HOD and GCH in the presence of large cardinals (including supercompact cardinals). In the remaining chapters, we use this coding to extend a variety of earlier results. In Chapter 4, we generalize theorems about the Ground Axiom to models with supercompact cardinals that satisfy GCH. In Chapter 5, we extend results in set theoretic geology to models that satisfy GCH. Finally, in Chapter 6, we use the coding to produce a model of the Wholeness Axiom, V = HOD and GCH.

Paul Ellis

Paul Ellis, Ph.D. 2009, Rutgers University, math genealogy | MathSciNet

Supervisor: Simon Thomas

Dissertation:  The classification problem for finite rank dimension groups

Abstract.  There has been much work done in the study of the Borel complexity of various naturally occurring classification problems. In particular, Hjorth and Thomas have shown that the Borel complexity of the classification problem for torsion-free abelian groups of finite rank increases strictly with rank. In this thesis, we extend this result to dimension groups of finite rank. As these groups are naturally characterized by Bratteli diagrams, we obtain a similar theorem for Bratteli diagrams. We also obtain a similar result for a class of countable simple locally finite groups which are also characterized by Bratteli diagrams.

Scott Schneider

Scott Schneider, Ph.D. 2009, Rutgers University, math genealogy | MathSciNet

Supervisor: Simon Thomas

Dissertation:  Borel superrigidity for actions of low rank lattices

Abstract.  A major recent theme in Descriptive Set Theory has been the study of countable Borel equivalence relations on standard Borel spaces, including their structure under the partial ordering of Borel reducibility. We shall contribute to this study by proving Borel incomparability results for the orbit equivalence relations arising from Bernoulli, profinite, and linear actions of certain subgroups of $\text{PSL}_2(\mathbb{R})$. We employ the techniques and general strategy pioneered by Adams and Kechris, and develop purely Borel versions of cocycle superrigidity results arising in the dynamical theory of semisimple groups.

Specifically, using Zimmer’s cocycle superrigidity theorems, we will prove Borel superrigidity results for suitably chosen actions of groups of the form $\text{PSL}_2(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers inside a multi-quadratic number field. In particular, for suitable primes $p\neq q$, we prove that the orbit equivalence relations arising from the natural actions of $\text{PSL}_2(\mathbb{Z}[\sqrt{q}])$ on the $p$-adic projective lines are incomparable with respect to Borel reducibility as $p, q$ vary. Furthermore, we also obtain Borel non-reducibility results for orbit equivalence relations arising from Bernoulli actions of the groups $\text{PSL}_2(\mathcal{O})$. In particular, we show that if $E_p$ denotes the orbit equivalence relation arising from a nontrivial Bernoulli action of $\text{PSL}_2(\mathbb{Z}[\sqrt{p}])$, then $E_p$ and $E_q$ are incomparable with respect to Borel reducibility whenever $p \neq q$.

Sam Coskey

Sam Coskey, Ph.D. 2008, Rutgers University, math genealogy | MathSciNet

Supervisor: Simon Thomas

Dissertation:  Descriptive aspects of torsion-free abelian groups

Abstract.  In recent years, a major theme in descriptive set theory has been the study of the Borel complexity of naturally occurring classification problems. For example, Hjorth and Thomas have shown that the Borel complexity of the isomorphism problem for the torsion-free abelian groups of rank $n$ increases strictly with the rank $n$. In this thesis, we present some new applications of the theory of countable Borel equivalence relations to various classification problems for the $p$-local torsion-free abelian groups of finite rank. Our main result is that when $n\geq 3$, the isomorphism and quasi-isomorphism problems for the $p$-local torsion-free abelian groups of rank $n$ have incomparable Borel complexities. (Here two abelian groups $A$ and $B$ are said to be quasi-isomorphic if $A$ is abstractly commensurable with $B$.) We also introduce a new invariant, the divisible rank, for the class of $p$-local torsion-free abelian groups of finite rank; and we prove that if $n\geq 3$ and $1 \leq k\leq n − 1$, then the isomorphism problems for the $p$-local torsion-free abelian groups of rank $n$ and divisible rank $k$ have incomparable Borel complexities as $k$ varies. Our proofs rely on the framework developed by Adams and Kechris, whereby cocycle superrigidity results from measurable group theory are applied in the purely Borel setting. In particular, we make use of the recent cocycle superrigidity theorem, due to Ioana, for free ergodic profinite actions of Kazhdan groups.   More


Joost WinterJoost Winter, M.Sc. 2007, Universiteit van Amsterdam

Supervisor: Benedikt Löwe

M.Sc. thesis:  Space compexity in infinite time Turing machines   pdf



Can BaskentCan Baskent, M.Sc. 2007, Universiteit van Amsterdam

Supervisor: Benedikt Löwe

M.Sc. Thesis: Topics in subset space logic


Yurii Khomski
Yurii Khomskii, M.Sc. 2007, Universiteit van Amsterdam

Supervisor: Benedikt Löwe

M.Sc. Thesis:  Regularity properties and determinacy



Erez ShochatErez Shochat, Ph.D. 2006, CUNY Graduate Center, math genealogy | MathSciNet

Supervisor:  Roman Kossak

Dissertation:  Countable short recursively saturated models of arithemtic

Abstract.  Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic.  Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated.  We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated.


Federico Marulanda ReyFederico Marulanda Rey, Ph.D. 2007, Columbia University, DBLP | Proquest | Google Books

Supervisor:  Haim Gaifman    (I was the outside reader)

Dissertation:  Contradiction, Paraconsistency, and Dialetheism

Abstract. The deductive closure of a set of sentences is trivial, i.e., it includes every well-formed sentence, if this set contains a contradiction and the consequence relation employed is either classical or intuitionistic. Over the past few decades, a number of paraconsistent logics, or logics specifically designed not to trivialize inconsistent theories, have been developed. The present work investigates philosophical issues arising from the development of paraconsistent formal systems. In the introductory chapter, as well as on a chapter that extracts learnings from Wittgenstein’s career-long preoccupation with contradiction, I endeavor to determine just what is the problem with contradictions, as they arise in both natural and formal languages. I then consider in detail two kinds of paraconsistent logic: their formal characteristics, the motivation for their formulation, their possible applications, and objections that may be raised against them. Special attention is devoted to a logical system that deliberately permits the evaluation of certain contradictions as being true, as well as to the attendant philosophical position, known as dialetheism, according to which there are, in fact, true contradictions. I raise a number of objections to this strong (and resilient) form of paraconsisteney, which, taken together, constitute a rebuttal of the view, thus carrying out a task that a number of authors have signaled as pressing, but which has not so far been undertaken in detail in the literature.


Ivan Welty, Ph.D. 2006, Columbia University, Philpapers | Google Books

Supervisor:  Haim Gaifman

Dissertation:  Frege Against Hilbert on the Foundations of Geometry

Abstract. This dissertation is a close study of the Frege-Hilbert dispute over the foundations of geometry. The dispute has been the subject of active debate recently, with opinion divided as to the merits of Frege’s position. In this dissertation I aim at a comprehensive assessment of Frege’s position, its motivations, and its major consequences. I find that: (1) Frege’s objections to Hilbert’s Foundations of Geometry do not represent a mere misunderstanding of Hilbert’s work, but stem from considerations of serious philosophical interest; (2) The same considerations that motivated Frege’s objections suggest a conception of geometry—and a reading of the history of geometry—radically different from Hilbert’s; (3) That conception of geometry—and reading of the history of geometry—are not obviously wrong, and indeed merit further investigation; (4) Part of Frege’s objection to Hilbert’s Foundations is that he gives no philosophical analysis of geometry, analogous to Frege’s analysis of number in Foundations of Arithmetic; (5) The basic framework for such an analysis can be found in Frege’s philosophical work, although it is far from obvious whether and how it can be carried through. The principal contributions of this dissertation lie in its clarification of the import of the Frege-Hilbert dispute for our understanding of the history of geometry, in particular the emergence of non-Euclidean and projective geometries; in its clarification of Frege’s objections to Hilbert’s independence proofs; and in its outline of a Fregean analysis of geometry, analogous to the analysis of number in Foundations of Arithmetic.

Sidney RafferSidney Raffer, Ph.D. 1999, CUNY Graduate Center

Supervisor:  Roman Kossak

math genealogy | MathSciNet

Dissertation: Some Diophantine properties of ordered polynomial rings