# Paul K. Gorbow, PhD 2018, University of Gothenburg

Paul K. Gorbow successfully defended his dissertation, “Self-similarity in the foundations” on June 14, 2018 at the University of Gothenburg in the Department of Philosophy, Linguistics and Theory of Science, under the supervision of Ali Enayat, with Peter LeFanu Lumsdaine and Zachiri McKenzie serving as secondary supervisors.  The defense opponent was Roman Kossak, with a dissertation committee consisting of Jon Henrik Forssell, Joel David Hamkins (myself) and Vera Koponen, chaired by Fredrik Engström. Congratulations!

Paul K. Gorbow, “Self-similarity in the foundations,” PhD dissertation for the University of Gothenburg, Acta Philosophica Gothoburgensia 32, June 2018. (arxiv:1806.11310)

Abstract. This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory.

The first part of the work on models of set theory consists in establishing a refined version of Friedman’s theorem on the existence of embeddings between countable non-standard models of a fragment of ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal the original model. The second part of the work on set theory consists in combining these two results into a technical machinery, yielding several results about non-standard models of set theory relating such notions as self-embeddings, their sets of fixed points, strong rank-cuts, and set theories of different strengths.

The work in foundational category theory consists in the formulation of a novel algebraic set theory which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic or classical NF, with or without atoms. A key axiom of this theory expresses that its structures have an endofunctor with natural properties.

In the Swedish style of dissertation defense, the opponent (in this case Roman Kossak) summarizes the dissertation, placing it in a broader context, and then challenges various parts of it, probing the candidate’s expertise in an extended discussion. What a pleasure it was to see this.  After this, there is a broader discussion, in which the committee is also involved.

# Kameryn J. Williams, PhD 2018, CUNY Graduate Center

Kameryn J. Williams successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 6th, 2018, earning his Ph.D. degree in May 2018. He has accepted a position in mathematics at the University of Hawaii, to begin Fall 2018.

What a pleasure it was to work with Kameryn, an extremely talented mathematician with wide interests and huge promise.

Kameryn J. Williams, The Structure of Models of Second-order Set Theories,  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2018. arXiv:1804.09526.

Abstract. This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recurions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or $\Pi^1_1$-CA—do not have least transitive models, while weaker theories—from GBC to GBC + ETR${}_{\text{Ord}}$—do have least transitive models.

In addition to his dissertation work and the research currently arising out of it, Kameryn has undertaken a number of collaborations with various international research efforts, including the following:

• He is a co-author on The exact strength of the class forcing theorem.
• V. Gitman, J. D. Hamkins, P. Holy, P. Schlicht, and K. Williams, “The exact strength of the class forcing theorem,” ArXiv e-prints, 2017. (manuscript under review)
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• He is co-author on a current joint project with Miha Habič, myself, Daniel Klausner and Jonathan Verner concerning the nonamalgamation phenomenon in the generic multiverse of a countable model of set theory.
• He is co-author on a current joint project with myself and Philip Welch concerning the universal $\Sigma_1$-definable finite sequence, an analogue of the universal finite set, but for the constructible universe.

# Corey Switzer, The Cichoń Diagram for Degrees of Relative Constructibility

My student Corey Switzer has just completed a paper:

Corey Switzer, The Cichoń Diagram for Degrees of Relative Constructibility, ArXiv e-print:1801.06497.

Abstract. Following a line of research initiated in Brendle/Brooke-Taylor/Ng/Nies, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cichoń diagram for cardinal characteristics of the continuum. I show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application, I define and investigate the Cichoń Diagram for degrees of constructibility relative to a fixed inner model W. Many analogies hold with the classical theory as well as some surprising differences. Along the way I introduce a new axiom stating, roughly, that the constructibility diagram is as complex as possible.

This interesting paper concerns a generalization of the Cichoń diagram to arbitrary reducibility notions, focussing on the case of the constructibility degrees, or somewhat more generally, relative constructibility $\leq_W$ over a fixed inner model $W$.

The classes are defined by an abstract generalization of the ideas underlying the familiar cardinal characteristics and the classical Cichoń diagram. Namely, $B_{\leq}(R)$ is the set of reals that $\leq$-build a witness for the fact that the reals of $W$ are small with respect to the relation $R$, that is, an $R$-bound for the reals of $W$; and $D_{\leq}(R)$ is the set of reals that $\leq$-build a witness for the fact that the reals of $W$ are not big with respect to the relation $R$, that is, a real that is not $R$-dominated by any real in $W$.

These classes fit together in a way that forms a robust analogy with the classical Cichoń diagram.

In his paper, Corey proves that the diagram is complete with respect to the inclusions indicated, by analyzing the nature of the diagram in various forcing extensions of $W$, such as the following.

In the end, he shows that in a suitable (proper) forcing extension, one can achieve all the separations simultaneously.

Indeed, the assertion that all separations are attained can be taken as a set-theoretic principle or axiom of its own, the complete Cichoń diagram assertion CD. He proves, for example, that CD is a consequence of the maximality principle.

# Kaethe Lynn Bruesselbach Minden, PhD 2017, CUNY Graduate Center

Kaethe Lynn Bruesselbach Minden successfully defended her dissertation on April 7, 2017 at the CUNY Graduate Center, under the supervision of Professor Gunter Fuchs. I was a member of the dissertation committee, along with Arthur Apter.

Her defense was impressive!  She was a master of the entire research area, ready at hand with the technical details to support her account of any topic that arose.

Kaethe Minden, “On Subcomplete Forcing,” Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2017. (arxiv/1705.00386)

Abstract. I survey an array of topics in set theory and their interaction with, or in the context of, a novel class of forcing notions: subcomplete forcing. Subcomplete forcing notions satisfy some desirable qualities; for example they don’t add any new reals to the model, and they admit an iteration theorem. While it is straightforward to show that any forcing notion which is countably closed is also subcomplete, it turns out that other well-known, more subtle forcing notions like Prikry forcing and Namba forcing are also subcomplete. Subcompleteness was originally defined by Ronald Björn Jensen around 2009. Jensen’s writings make up the vast majority of the literature on the subject. Indeed, the definition in and of itself is daunting. I have attempted to make the subject more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class.

It is well-known that countably closed forcings cannot add branches through $\omega_1$-trees. I look at the interaction between subcomplete forcing and $\omega_1$-trees. It turns out that sub-complete forcing also does not add cofinal branches to $\omega_1$-trees. I show that a myriad of other properties of trees of height $\omega_1$ as explored in [FH09] are preserved by subcomplete forcing; for example, I show that the unique branch property of Suslin trees is preserved by subcomplete forcing.

Another topic I explored is the Maximality Principle ($\text{MP}$). Following in the footsteps of Hamkins [Ham03], Leibman [Lei], and Fuchs [Fuc08], [Fuc09], I examine the subcomplete maximality principle. In order to elucidate the ways in which subcomplete forcing generalizes the notion of countably closed forcing, I compare the countably closed maximality principle ($\text{MP}_{<\omega_1\text{-closed}}$) to the subcomplete maximality principle ($\text{MP}_{sc}$). Again, since countably closed forcing is subcomplete, this is a natural question to ask. I was able to show that many of the results about $\text{MP}_{<\omega_1\text{-closed}}$ also hold for $\text{MP}_{sc}$; for example, the boldface appropriate notion of $\text{MP}_{sc}$ is equiconsistent with a fully reflecting cardinal. However, it is not the case that there are direct implications between the subcomplete and countably closed maximality principles.

Another forcing principle explored in my thesis is the Resurrection Axiom ($\text{RA}$). Hamkins and Johnstone [HJ14a] defined the resurrection axiom only relative to $H_{\mathfrak{c}}$, and focus mainly on the resurrection axiom for proper forcing. They also show the equiconsistency of various resurrection axioms with an uplifting cardinal. I argue that the subcomplete resurrection axiom should naturally be considered relative to $H_{\omega_2}$, and showed that the subcomplete resurrection axiom is equiconsistent with an uplifting cardinal.

A question reasonable to ask about any class of forcings is whether or not the resurrection axiom and the maximality principle can consistently both hold for that class. I originally had this question about the full principles, not restricted to any class, but in my thesis it was appropriate to look at the question for subcomplete forcing. I answer the question positively for subcomplete forcing using a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals needed to force the principles separately. I show that the boldface versions of $\text{MP}_{sc}+\text{RA}_{sc}$ both holding is equiconsistent with the existence of a strongly uplifting fully reflecting cardinal. While Jensen [Jen14] shows that Prikry forcing is subcomplete, I long suspected that many variants of Prikry forcing which have a kind of genericity criterion are also subcomplete. After much work I managed to show that a variant of Prikry forcing known as Diagonal Prikry Forcing is subcomplete, giving another example of subcomplete forcing to add to the list.

Kaethe has taken up a faculty position at Marlboro College in Vermont.

# Miha E. Habič, PhD 2017, CUNY Graduate Center

Miha E. Habič successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 7th, 2017, earning his Ph.D. degree in May 2017.

It was truly a pleasure to work with Miha, who is an outstanding young mathematician with enormous promise. I shall look forward to seeing his continuing work.

Miha E. Habič, “Joint Laver diamonds and grounded forcing axioms,”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2017 (arxiv:1705.04422).

Abstract. In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for $\kappa$ is joint if for any sequence of targets there is a single elementary embedding $j$ with critical point $\kappa$ such that each Laver diamond guesses its respective target via $j$. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for $\kappa$ yields a joint sequence of length $\kappa$, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of $\theta$-strong cardinals where, for certain $\theta$, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary $\diamondsuit_\kappa$-sequences on any regular cardinal $\kappa$. The main result concerning these shows that there is no separation according to length and a single $\diamondsuit_\kappa$-sequence yields joint families of all possible lengths.

In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin’s axiom. This grounded Martin’s axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin’s axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin’s axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin’s axiom itself.

Miha will shortly begin a post-doctoral research position at Charles University in Prague.

# Regula Krapf, Ph.D. 2017, University of Bonn

Regula Krapf successfully defended her PhD dissertation January 12, 2017 at the University of Bonn, with a dissertation entitled, “Class forcing and second-order arithmetic.”  I was a member of the dissertation examining committee. Peter Koepke was the dissertation supervisor.

Regula Krapf, Class forcing and second-order arithmetic, dissertation 2017, University of Bonn. (Slides)

Abstract. We provide a framework in a generalization of Gödel-Bernays set theory for performing class forcing. The forcing theorem states that the forcing relation is a (definable) class in the ground model (definability lemma) and that every statement that holds in a class-generic extension is forced by a condition in the generic filter (truth lemma). We prove both positive and negative results concerning the forcing theorem. On the one hand, we show that the definability lemma for one atomic formula implies the forcing theorem for all formulae in the language of set theory to hold. Furthermore, we introduce several properties which entail the forcing theorem. On the other hand, we give both counterexamples to the definability lemma and the truth lemma. In set forcing, the forcing theorem can be proved for all forcing notions by constructing a unique Boolean completion. We show that in class forcing the existence of a Boolean completion is essentially equivalent to the forcing theorem and, moreover, Boolean completions need not be unique.

The notion of pretameness was introduced to characterize those forcing notions which preserve the axiom scheme of replacement. We present several new characterizations of pretameness in terms of the forcing theorem, the preservation of separation, the existence of nice names for sets of ordinals and several other properties. Moreover, for each of the aforementioned properties we provide a corresponding characterization of the Ord-chain condition.

Finally, we prove two equiconsistency results which compare models of ZFC (with large cardinal properties) and models of second-order arithmetic with topological regularity properties (and determinacy hypotheses). We apply our previous results on class forcing to show that many important arboreal forcing notions preserve the $\Pi^1_1$-perfect set property over models of second-order arithmetic and also give an example of a forcing notion which implies the $\Pi^1_1$-perfect set property to fail in the generic extension.

Regula has now taken up a faculty position at the University of Koblenz.

# Jacob Davis, PhD 2016, Carnegie Mellon University

Jacob Davis successfully defended his dissertation, “Universal Graphs at $\aleph_{\omega_1+1}$ and Set-theoretic Geology,” at Carnegie Mellon University on April 29, 2016, under the supervision of James Cummings. I was on the dissertation committee (participating via Google Hangouts), along with Ernest Schimmerling and Clinton Conley.

The thesis consisted of two main parts. In the first half, starting from a model of ZFC with a supercompact cardinal, Jacob constructed a model in which $2^{\aleph_{\omega_1}} = 2^{\aleph_{\omega_1+1}} = \aleph_{\omega_1+3}$ and in which there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.  The same technique works with any uncountable cardinal in place of $\omega_1$.  In the second half, Jacob proved a variety of results in the area of set-theoretic geology, including several instances of the downward directed grounds hypothesis, including an analysis of the chain condition of the resulting ground models.

# Burak Kaya, Ph.D. March 2016

Burak Kaya successfully defended his dissertation, “Cantor minimal systems from a descriptive perspective,” on March 24, 2016, earning his Ph.D. degree at Rutgers University under the supervision of Simon Thomas. The dissertation committee consisted of Simon Thomas, Gregory Cherlin, Grigor Sargsyan and myself, as the outside member.

The defense was very nice, with an extremely clear account of the main results, and the question session included a philosophical discussion on various matters connected with the dissertation, including the principle attributed to Gao that any collection of mathematical structures that has a natural Borel representation has a unique such representation up to Borel isomorphism, a principle that was presented as a Borel-equivalence-relation-theory analogue of the Church-Turing thesis.

Abstract.  In recent years, the study of the Borel complexity of naturally occurring classification problems has been a major focus in descriptive set theory. This thesis is a contribution to the project of analyzing the Borel complexity of the topological conjugacy relation on various Cantor minimal systems.

We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $\newcommand\d{\Delta^+_{\mathbb{R}}}\d$. As a byproduct of our analysis, we also show that $\d$ is a lower bound for the topological conjugacy relation on Cantor minimal systems.

The other main result of this thesis concerns the topological conjugacy relation on Toeplitz subshifts. We prove that the topological conjugacy relation on Toeplitz subshifts with separated holes is a hyperfinite Borel equivalence relation. This result provides a partial affirmative answer to a question asked by Sabok and Tsankov.

As pointed Cantor minimal systems are represented by properly ordered Bratteli diagrams, we also establish that the Borel complexity of equivalence of properly ordered Bratteli diagrams is $\d$.

# Giorgio Audrito, PhD 2016, University of Torino

Dr. Giorgio Audrito has successfully defended his dissertation, “Generic large cardinals and absoluteness,” at the University of Torino under the supervision of Matteo Viale.

The dissertation Examing Board consisted of myself (serving as Presidente), Alessandro Andretta and Sean Cox.  The defense took place March 2, 2016.

The dissertation was impressive, introducing (in joint work with Matteo Viale) the iterated resurrection axioms $\text{RA}_\alpha(\Gamma)$ for a forcing class $\Gamma$, which extend the idea of the resurrection axioms from my work with Thomas Johnstone, The resurrection axioms and uplifting cardinals, by making successive extensions of the same type, forming the resurrection game, and insisting that that the resurrection player have a winning strategy with game value $\alpha$. A similar iterative game idea underlies the $(\alpha)$-uplifting cardinals, from which the consistency of the iterated resurrection axioms can be proved. A final chapter of the dissertation (joint with Silvia Steila), develops the notion of $C$-systems of filters, generalizing the more familiar concepts of extenders and towers.

# Set Theory Day at the CUNY Graduate Center, March 11, 2016

Vika Gitman, Roman Kossak and Miha Habič have been very kind to organize what they have called Set Theory Day, to be held Friday March 11 at the CUNY Graduate Center in celebration of my 50th birthday. This will be an informal conference focussing on the research work of my various PhD graduate students, and all the lectures will be given by those who were or are currently a PhD student of mine. It will be great! I am very pleased to count among my former students many who have now become mathematical research colleagues and co-authors of mine, and I am looking forward to hearing the latest. If you want to hear what is going on with infinity, then please join us March 11 at the CUNY Graduate Center!

(The poster was designed by my student Erin Carmody, who graduated last year and now has a position at Nebraska Wesleyan.)

# Erin Carmody

Erin Carmody successfully defended her dissertation under my supervision at the CUNY Graduate Center on April 24, 2015, and she earned her Ph.D. degree in May, 2015. Her dissertation follows the theme of killing them softly, proving many theorems of the form: given $\kappa$ with large cardinal property $A$, there is a forcing extension in which $\kappa$ no longer has property $A$, but still has large cardinal property $B$, which is very slightly weaker than $A$. Thus, she aims to enact very precise reductions in large cardinal strength of a given cardinal or class of large cardinals. In addition, as a part of the project, she developed transfinite meta-ordinal extensions of the degrees of hyper-inaccessibility and hyper-Mahloness, giving notions such as $(\Omega^{\omega^2+5}+\Omega^3\cdot\omega_1^2+\Omega+2)$-inaccessible among others.

Erin Carmody, “Forcing to change large cardinal strength,”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2015.  ar$\chi$iv | PDF

Erin has accepted a professorship at Nebreska Wesleyan University for.the 2015-16 academic year.

Erin is also an accomplished artist, who has had art shows of her work in New York, and she has pieces for sale. Much of her work has an abstract or mathematical aspect, while some pieces exhibit a more emotional or personal nature. My wife and I have two of Erin’s paintings in our collection:

# Norman Lewis Perlmutter

Norman Lewis Perlmutter successfully defended his dissertation under my supervision and will earn his Ph.D. at the CUNY Graduate Center in May, 2013.  His dissertation consists of two parts.  The first chapter arose from the observation that while direct limits of large cardinal embeddings and other embeddings between models of set theory are pervasive in the subject, there is comparatively little study of inverse limits of systems of such embeddings.  After such an inverse system had arisen in Norman’s joint work on Generalizations of the Kunen inconsistency, he mounted a thorough investigation of the fundamental theory of these inverse limits. In chapter two, he investigated the large cardinal hierarchy in the vicinity of the high-jump cardinals.  During this investigation, he ended up refuting the existence of what are now called the excessively hypercompact cardinals, which had appeared in several published articles.  Previous applications of that notion can be made with a weaker notion, what is now called a hypercompact cardinal.

Norman Lewis Perlmutter, “Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2013.

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 fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, 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 in both directions by forcing under fairly general assumptions. 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 $\sup\{\ j(f)(\kappa) \mid f: \kappa \to \kappa\ \}$.  Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.

# Brent Cody

Brent Cody earned his Ph.D. under my supervision at the CUNY Graduate Center in June, 2012.  Brent’s dissertation work began with the question of finding the exact consistency strength of the GCH failing at a cardinal $\theta$, when $\kappa$ is $\theta$-supercompact.  The answer turned out to be a $\theta$-supercompact cardinal that was also $\theta^{++}$-tall.  After this, he quickly dispatched more general instances of what he termed the Levinski property for a variety of other large cardinals, advancing his work towards a general investigation of the Easton theorem phenomenon in the large cardinal context, which he is now undertaking.  Brent held a post-doctoral position at the Fields Institute in Toronto, afterwards taking up a position at the University of Prince Edward Island.  He is now at Virginia Commonwealth University.

Brent Cody

Brent Cody, “Some Results on Large Cardinals and the Continuum Function,” Ph.D. dissertation for The Graduate Center of the City University of New York, June, 2012.

Abstract.  Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F”\delta\subseteq\delta$ and GCH holds, then there is a cofinality preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal $\gamma<\delta$, and in which $\delta$ remains Woodin.

I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$ is equiconsistent with the existence of a cardinal $\kappa$ that is $\lambda$-supercompact and $\lambda^{++}$-tall.

I generalize a theorem on measurable cardinals due to Levinski, which says that given a measurable cardinal, there is a forcing extension preserving the measurability of $\kappa$ in which $\kappa$ is the least regular cardinal at which GCH holds. Indeed, I show that Levinski’s result can be extended to many other large cardinal contexts. This work paves the way for many additional results, analogous to the results stated above for Woodin cardinals and partially supercompact cardinals.

# 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

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

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 $\Delta^1_1$.