# Welcome to New York Logic

Welcome to New York Logic!

Apr 20 Monday Apr 21 Tuesday Apr 22 Wednesday Apr 23 Thursday Apr 24 Friday Apr 25 Saturday Apr 26 Sunday

Models of PAApr 22, 20154:50 pm
Set theory seminarApr 24, 201510:00 am

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## Upcoming talks and events:

Computational Logic SeminarTuesday, April 21, 20152:00 pmGraduate Center, rm. 6421

# A general completeness theorem relating sequents and bivaluations

Federal University of Rio de Janeiro, Brazil

In this talk I will present a theorem justifying an intuitive semantical interpretation of rules of sequent proof systems. This theorem is based on the one hand on a generalization of Lindenbuam maximalization theorem and on the other hand on a general semantical theory of bivaluations, not necessarily truth-functional, originally developed by Newton da Costa for paraconsistent systems. I will show that this theorem gives a better understanding of sequent calculus and that it can fruitfully be applied to a wide class of logical systems providing instantaneous completeness results. This is a typical example of work in the line of the universal logic project, of which I will say a few words.

Reference: Logica Universalis, Special Double Issue – Scope of Logic Theorems. Volume 8, Issue 3-4, December 2014.

Models of PAWednesday, April 22, 20154:50 pmGC 6300

# A perfectly generic talk

We will look at perfect generics for countable models of PA. Our goal will be to prove the following theorems: 1. Any countable collection of inductive subsets of a countable model are definable from a single generic. 2. Any countable model has minimally undefinable generics.

Set theory seminarFriday, April 24, 201510:00 amGC 6417

# When is a Boolean ultrapower an ultrapower?

CUNY Borough of Manhattan Community College

The Boolean ultrapower construction is a natural generalization of the classical ultrapower construction, but the Boolean ultrapower uses an ultrafilter on a complete Boolean algebra instead of a set. It was initially unknown as to whether in ZFC there exists a Boolean ultrapower which is not always isomorphic to a classical ultrapower. This problem was resolved in 1976 by Bernd and Sabine Koppelberg who constructed a Boolean ultrapower which is not an ultrapower. On the other hand, there does not seem to be any reference in the mathematical literature to atomless Boolean ultrapowers which are isomorphic to classical ultrapowers.

We shall first generalize the notion of a Boolean ultrapower to the notion of a BPA-ultrapower which is in a sense the most general ultrapower construction. Then by applying a result of Joel David Hamkins which characterizes the Boolean ultrapowers which are classical ultrapowers, we shall investigate examples of Boolean ultrapowers which are not classical ultrapowers as well as Boolean ultrapowers which are classical ultrapowers. For instance, I claim that under GCH every complete atomless Boolean algebra has an ultrafilter which gives rise to a Boolean ultrapower which is not a classical ultrapower. On the other hand, using the Keisler-Shelah isomorphism theorem, we may construct Boolean ultrapowers in ZFC on a fairly general class of Boolean algebras which are classical ultrapowers.

Kolchin seminar in Differential AlgebraFriday, April 24, 201510:15 amGC 5382

# A Differential Algebra Sampler

University of Pennsylvania

We discuss several problems involving differential algebraic varieties and ideals in differential polynomial rings. The first one is the completeness of projective differential varieties. We consider examples showing the failure to generalize (even in the finite-rank case) of the classical “fundamental theorem of elimination theory”. We also treat identification of complete differential varieties and a connection to the differential catenary problem. We finish by examining what proof-theoretic techniques have to say about the constructive content of results such as the Ritt-Raudenbush basis theorem and differential Nullstellensatz. Our remarks include work with James Freitag and Omar León-Sánchez as well as ongoing work with Henry Towsner.

Model theory seminarFriday, April 24, 201512:30 pmGC 6417

# Fermat’s Last Theorem and Catalan’s conjecture in weak exponential arithmetics

Charles University

This is a joint work with Vitezslav Kala.

Wiles’s proof of Fermat’s Last Theorem (FLT) has stimulated a lively discussion on how much is actually needed for the proof.
Despite the fact that the original proof uses set-theoretical assumptions unprovable in Zermelo-Fraenkel set theory with axiom of choice (ZFC) (namely, the existence of Grothendieck universes), it is widely believed that
certainly much less than ZFC is used in principle, probably nothing beyond Peano arithmetic, and perhaps much less than that.

I will start with a brief summary of existing positive and negative results on provability of FLT in various arithmetical theories.

In this talk, we will consider structures and theories in the language L=(0,1,+,x,<,e), where the symbol e is intended for a (partial or total) binary exponential. We show that Fermat's Last Theorem for e (i.e. the statement "e(a,n)+e(b,n)=e(c,n) has no non-zero solution for n>2″) is not provable in the L-theory Th(N)+Exp, where Th(N) stands for the complete theory of the standard model N=(N,0,1,+,x,<) and Exp is a natural set of axioms for e (consisting mostly of elementary identities).

On the other hand, under the assumption of ABC conjecture (in the standard model), we show that the Catalan conjecture for e is provable in Th(N)+Exp (even in a weaker theory). This gives an interesting separation of strengths of these two diophantine problems.

Finally, we also show that Fermat’s Last Theorem for e is provable (again, under the assumption of ABC in N) in Th(N)+Exp +”coprimality for e”.

Set theory seminarFriday, April 24, 20152:00 pmGC 6417

# Dissertation Defense: Force to change large cardinals

This will be the dissertation defense of the speaker.  There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.

Set theory seminarFriday, May 1, 201510:00 amGC 6417

# Diamond* Coding

CUNY New York City College of Technology

Coding information into the structure of the universe is a forcing technique with many applications in set theory. To carry out it out, we a need a property that: i) can be easily switched on or off at (e.g.) each regular cardinal in turn, and ii) is robust with regards both to small and to highly-closed forcing. GCH coding, controlling the success or failure of the GCH at each cardinal in turn, is the most widely used, and for good reason: there are simple forcings that turn it on and off, and it is easily seen to be unaffected by small or highly-closed forcing. However, it does have limitations – most obviously, GCH coding is of necessity incompatible with the GCH itself. In this talk I will present an alternative coding using the property Diamond*, a variant of the classic Diamond. I will discuss Diamond* and demonstrate that it satisfies the requirements for coding while preserving the GCH.

Although the basic techniques for controlling Diamond* have been known for some time, to my knowledge the first use of Diamond* as a coding axiom was by Andrew Brooke-Taylor in his work on definable well-orders of the universe. I will follow the excellent exposition presented in his dissertation.

Kolchin seminar in Differential AlgebraFriday, May 1, 201510:15 amGC 5382

# A Differential Hensel’s Lemma for Local Algebras

McMaster University

We will discuss a differential version of the classical Hensel’s lemma on lifting solutions from the residue field (working on a local artinian differential algebra over a differentially closed field). We will also talk about some generalizations; for example, one can remove the locality hypothesis by assuming finite dimensionality. If time permits, I will give an easy application on extensions of generalized Hasse-Schmidt operators. This is joint work with Rahim Moosa.

CUNY Logic WorkshopFriday, May 1, 20152:00 pmGC 6417

# A Local Characterization of VC-minimality

University of Wisconsin

(Work joint with Vincent Guingona.) I’ll talk about a problem in the intersection of computable model theory and classical model theory. The notion of VC-minimality, though intriguing, has proven very difficult to work with. It has even been difficult to check whether familiar examples are VC-minimal. This has led model theorists (chiefly my co-author) to ask whether there was a “local” (read “simpler”) characterization of VC-minimality. I suggested a computable model theoretic version of this question in terms of the index set of VC-minimality. We answered this question by giving a local characterization of VC-minimality, and we showed that VC-minimality is a Π04-complete notion.

Model theory seminarFriday, May 8, 201512:30 pmGC 6417

# Title TBA

George Washington University
Model theory seminarFriday, May 15, 201512:30 pmGC 6417

# Actions on sets of Morley rank $2$

Hamilton College

Recently, Borovik and Cherlin initiated a broad study of permutation groups of finite Morley rank with a key topic being high degrees of generic transitivity. One of the main problems that they pose is to show that there is a natural upper bound on the degree of generic transitivity that depends only upon the rank of the set being acted on. Specifically, the problem is to show that the only groups of finite Morley rank with a generically $(n+2)$-transitive action on a set of rank $n$ are those of the form ${PGL}_{n+1}$. A solution when $n=1$, due to Hrushovski, has been known for a few decades as in this case the set is strongly minimal. In this talk, I will present recent work, joint with Tuna Altinel, addressing the case of $n=2$. The analysis of these actions makes considerable use of the structure of groups of small rank, and as such, I will also discuss some new results on groups of Morley rank $4$.

Kolchin seminar in Differential AlgebraFriday, May 22, 201510:15 amGC 5382

# Title TBA

University of Illinois at Urbana-Champaign
Model theory seminarFriday, May 22, 201512:30 pmGC 6417

# TBA

University of California, Los Angeles