Category Archives: Students

Davide Leonessi, MSc MFoCS, Oxford, September 2021

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Mr. Davide Leonessi successfully defended his dissertation for the Masters of Science degree in Mathematics and Foundations of Computer Science, entitled “Transfinite game values in infinite games,” on 15 September 2021. Davide earned a distinction for his thesis, an outstanding result.

Davide Leonessi | Google scholar | Dissertation | arXiv

Abstract. The object of this study are countably infinite games with perfect information that allow players to choose among arbitrarily many moves in a turn; in particular, we focus on the generalisations of the finite board games of Hex and Draughts.

In Chapter 1 we develop the theory of transfinite ordinal game values for open infinite games following [Evans-Hamkins 2014], and we focus on the properties of the omega one, that is the supremum of the possible game values, of classes of open games; we moreover design the class of climbing-through-$T$ games as a tool to study the omega one of given game classes.

The original contributions of this research are presented in the following two chapters.

In Chapter 2 we prove classical results about finite Hex and present Infinite Hex, a well-defined infinite generalisation of Hex.

We then introduce the class of stone-placing games, which captures the key features of Infinite Hex and further generalises the class of positional games already studied in the literature within the finite setting of Combinatorial Game Theory.

The main result of this research is the characterization of open stone-placing games in terms of the property of essential locality, which leads to the conclusion that the omega one of any class of open stone-placing games is at most $\omega$. In particular, we obtain that the class of open games of Infinite Hex has the smallest infinite omega one, that is $\omega_1^{\rm Hex}=\omega$.

In Chapter 3 we show a dual result; we define the class of games of Infinite Draughts and explicitly construct open games of arbitrarily high game value with the tools of Chapter 1, concluding that the omega one of the class of open games of Infinite Draughts is as high as possible, that is $\omega_1^{\rm Draughts}=\omega_1$.

The full dissertation is available:

Transfinite-game-values-in-infinite-gamesDownload

Corey Bacal Switzer, PhD 2020, CUNY Graduate Center

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Dr. Corey Bacal Switzer successfully defended his PhD dissertation, entitled “Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH,” on 31 July 2020, for the degree of PhD from The Graduate Center of the City University of New York. The dissertation was supervised jointly by myself and Gunter Fuchs.

Corey has now accepted a three-year post-doctoral research position at the University of Vienna, where he will be working with Vera Fischer.

Corey Bacal Switzer | arXiv.org | Google scholar | dissertation

Abstract. This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are four largely independent chapters, the first two of which consider alternative versions of the Cichoń diagram and the latter two consider forcing axioms compatible with CH . In the first chapter, I begin by introducing the notion of a reduction concept , generalizing various notions of reduction in the literature and show that for each such reduction there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model $W\models\text{ZFC}$.

In the second chapter, I study the space of functions $f:\omega^\omega\to\omega^\omega$ and introduce 18 new higher cardinal characteristics associated with this space. I prove that these can be organized into two diagrams of 6 and 12 cardinals respecitvely analogous to the Cichoń diagram on $\omega$. I then investigate their relation to cardinal invariants on ω and introduce several new forcing notions for proving consistent separations between the cardinals. The third chapter concerns Jensen’s subcomplete and subproper forcing. I generalize these notions to the (seemingly) larger classes of ∞-subcomplete and ∞-subproper. I show that both classes are (apparently) much more nicely behaved structurally than their non-∞-counterparts and iteration theorems are proved for both classes using Miyamoto’s nice iterations. Several preservation theorems are then presented. This includes the preservation of Souslin trees, the Sacks property, the Laver property, the property of being $\omega^\omega$-bounding and the property of not adding branches to a given $\omega_1$-tree along nice iterations of ∞-subproper forcing notions. As an application of these methods I produce many new models of the subcomplete forcing axiom, proving that it is consistent with a wide variety of behaviors on the reals and at the level of $\omega_1$.

The final chapter contrasts the flexibility of SCFA with Shelah’s dee-complete forcing and its associated axiom DCFA . Extending a well known result of Shelah, I show that if a tree of height $\omega_1$ with no branch can be embedded into an $\omega_1$-tree, possibly with branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails.

Paul K. Gorbow, PhD 2018, University of Gothenburg

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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!

University of Gothenburg profile |  ar$\chi$iv | Research Gate

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.