Oxford Set Theory Seminar

I am pleased to announce the founding of the Oxford Set Theory Seminar.

We shall focus on all aspects of set theory and the philosophy of set theory.

Topics will include forcing, large cardinals, models of set theory, set theory as a foundation, set-theoretic potentialism, cardinal characteristics of the continuum, second-order set theory and class theory, and much more.

Technical topics are completely fine. Speakers are encouraged to pick set-theoretic topics having some philosophical angle or aspect, although it is expected that this might sometimes be a background consideration, while at other times it will be a primary focus.

The seminar will last 60-90 minutes. Speakers are requested to prepare a one hour talk, and we expect a lively discussion with questions.

Trinity Term 2020

In Trinity term 2020, the seminar is organized by myself and Samuel Adam-Day. In light of the corona virus situation, we will be meeting online via Zoom for the foreseeable future.

For the Zoom access code, contact Samuel Adam-Day me@samadamday.com.

6 May 2020, 4 pm UK

Victoria Gitman, City University of New York

Elementary embeddings and smaller large cardinals

Abstract  A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

20 May 2020, 4 pm

Joel David Hamkins, Oxford

Bi-interpretation of weak set theories

Abstract. Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

27 May 2020, 4 pm

Ali Enayat, Gothenberg

Leibnizian and anti-Leibnizian motifs in set theory

Abstract. Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense:  LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles.  LM was further investigated in a 2004  paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory.  On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

17 June 2020, 4 pm

Corey Bacal Switzer, City University of New York

Some Set Theory of Kaufmann Models

Abstract. A Kaufmann model is an $\omega_1$-like, recursively saturated, rather classless model of PA. Such models were shown to exist by Kaufmann under the assumption that $\diamondsuit$ holds, and in ZFC by Shelah via an absoluteness argument involving strong logics. They are important in the theory of models of arithmetic notably because they show that many classic results about countable, recursively saturated models of arithmetic cannot be extended to uncountable models. They are also a particularly interesting example of set theoretic incompactness at $\omega_1$, similar to an Aronszajn tree.

In this talk we’ll look at several set theoretic issues relating to this class of models motivated by the seemingly naïve question of whether or not such models can be killed by forcing without collapsing $\omega_1$. Surprisingly the answer to this question turns out to be independent: under $\mathsf{MA}_{\aleph_1}$ no $\omega_1$-preserving forcing can destroy Kaufmann-ness whereas under $\diamondsuit$ there is a Kaufmann model $M$ and a Souslin tree $S$ so that forcing with $S$ adds a satisfaction class to $M$ (thus killing rather classlessness). The techniques involved in these proofs also yield another surprising side of Kaufmann models: it is independent of ZFC whether the class of Kaufmann models can be axiomatized in the logic $L_{\omega_1, \omega}(Q)$ where $Q$ is the quantifier “there exists uncountably many”. This is the logic used in Shelah’s aforementioned result, hence the interest in this level of expressive power.

The seminar talks appear in the compilation of math seminars at https://mathseminars.org/seminar/oxford-set-theory.

Drunk Science: Infinity, special guest, June 23, 2016

Drunk Science

I shall be special guest at Drunk Science: Infinity, an experimental comedy show in Brooklyn, during which three intoxicated comedians will compete to offer the best dissertation defense on the topic of my research.

The event will take place Thursday, June 23, 2016, (doors 7pm, show 8pm) at the Littlefield performance and art space, 622 Degraw Street between 3rd and 4th Avenue in Brooklyn. Tickets from $5.  (Get tickets now, since the shows often sell out.)

Update: What a riot it was! I really had a lot of fun.


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!

Set Theory Day 2016

Vika Gitman’s announcement of Set Theory Day |  Set Theory Day conference web page | My graduate students

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

A conference in honor of W. Hugh Woodin’s 60th birthday, March 2015

I am pleased to announce the upcoming conference at Harvard celebrating the 60th birthday of W. Hugh Woodin.  See the conference web site for more information. Click on the image below for a large-format poster.


Just do it? Barbara Gail Montero interviewed on The Philosopher's Zone

Barbara’s radio interview this week on Radio National:


Just do it?

November 3, 2013
BARBARA GAIL MONTERO interviewed by Joe Gelonesi along with Richard Menary on The Philosopher’s Zone.

Famed choreographer George Balanchine was reputed to have said, “don’t think, dear: just do”. The idea that champion performers switch off their brains to achieve their best has taken hold in popular imagination. Just do it promises an existential zone where real players hit the heights whilst the rest shuffle to the back of the pack. We explore Expert action, a philosophical football punted between those for automatic responses and those who hear the whirring cogs.  

→ go listen to `Just Do It

Barbara was previously interviewed on Leading Minds, with David Brendel.

Rubik's cube competition, CSI, November 14, 2013

Rubik's cube 2

Come and compete in the CSI Rubik’s cube competition!

November 14, 2013, College of Staten Island of CUNY, 1S-107, 2:30 pm.

Sponsored by MTH 339, and the CSI Math Club.

As a part of the undergraduate course in abstract algebra (MTH 339), which I am teaching this semester at the College of Staten Island, we shall hold a Rubik’s cube competition on November 14th.  In class, I have used the Rubik’s cube as a source of examples to explain various group-theoretic concepts, and I have encouraged the students to learn to solve the cube.  Several have now already mastered it, and there seems lately to be a lot of Rubik’s cube activity in the math department.  (I am giving extra credit for any student who can solve a scrambled cube in my office.)

Several students have learned how to solve the cube from the following video, which explains one of the layer-based solution methods:

Free New York Pizza!

The Competition.  On November 14, 2013, we will have the Rubik’s cube competition, with several rounds of competition, to see who can solve the cube the fastest.  Prizes will be awarded, and best of all, there will be free pizza!

Results Of the Competition

The event has now taken place. We had 15 competitors, from all around the College and beyond.  We organized two qualifying heats of 7 and 8 competitors, respectively, taking the top four from each qualtifying heat to form the quarterfinalist competitors. The top four of these formed the semifinalist competitors. And the top two of these headed off in the championship round.  The champion, Sam Obisanya, won all the rounds in which he competed, and his cube was a blaze of lightning color as he solved it.  Honorable mention goes especially to Oveen Joseph, who faced Sam in the championship round and who came out to the college from middle school I.S.72, where he is in the 7th grade, and also to Justin Mills, who had extremely fast times.


Itiel Cohen (CSI math major)

William George (CSI math major)

Oveen Joseph (middle school I.S.72, 7th grade)

Wing Yang Law (CSI math major)

Justin Mills (CSI psychology major)

Mike Siozios (CSI math major)

Sam Obisanya (CSI nursing major)

James Yap (CSI math major)


Oveen Joseph

Justin Mills

Sam Obisanya

James Yap

Championship round:

Oveen Joseph

Sam Obisanya

Final Champion:

 Sam Obisanya

Congratulations to our champion and to all the competitors.

Rubik's cube


MAMLS at Rutgers, October 6-7, 2012

The Fall 2012 MAMLS Meeting will take place at Rutgers University on October 6-7, 2012. The invited speakers include Clinton Conley, Andrew Marks, Antonio Montalban, Justin Moore, Saharon Shelah, Dima Sinapova and Anush Tserunyan.

The lectures will take place in Room 216 in Scott Hall on College Avenue Campus. For those of you who are coming by train, Scott Hall is a short walk from the train station.

For further information, visit:


Panel discussion on the unity and diversity of logic, New York, March 2012

As a part of the Spring 2012 Mid-Atlanatic Mathematical Logic Seminar, to be held March 9-10, 2012 at the CUNY Graduate Center, I shall participate in the following panel discussion.

Panel discussion: The unity and diversity of logic

Abstract.  The field of mathematical logic sometimes seems to be fracturing into ever-finer subdisciplines, with little connection between them, and many logicians now identify themselves by their specific subdiscipline.  On the other hand, certain new themes have appeared which tend to unify the diverse discoveries of the many subdisciplines.  This discussion will address these trends and ask whether one is likely to dominate the other in the long term.  Will logic remain a single field, or will it split into many unrelated branches?

The panelists will be Prof. Gregory Cherlin, myself, Prof. Rohit Parikh, and Prof. Jouko Väänänen, with the discussion moderated by Prof. Russell Miller. Questions and participation from the audience are encouraged.

As preparation for this panel discussion, please suggest points or topics that might brought up at the panel discussion, by posting suitable comments below.  Perhaps we’ll proceed with our own pre-discussion discussion here!