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 + sloth

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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 Minden defense

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 Habic

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

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.

Giorgio Audrito defense (small)

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.