This Week in Logic at CUNY:

– – – – Monday, Jan 30, 2012 – – – –

– – – – Tuesday, Jan 31, 2012 – – – –

Computational Logic Seminar

Room 4433. Time 2:00 – 4:00 PM, January 31

Speaker: Juoko Vaananen (Helsinki – IAS Princeton)

Title: The logic of dependence and independence

Abstract: In my talk I isolate a concept of dependence (“Dependence Logic”, Cambridge University Press, 2007) as well as a concept of independence and argue that these concepts, ubiquitous in science and humanities, but also in logic and computer science, can be taken as new atomic formulas in first order logic and be axiomatized like their cousin identity is axiomatized by the identity axioms. I will go on to show that while the resulting extension of first order logic is non-axiomatizable, there are canonical natural deduction rules which completely axiomatize first order consequences of sentences in this logic. On finite models the resulting logic gives a new analysis of NP. I will give a review of recent results on the expressive power of dependence and independence logics, as well as some applications.

– – – – Wednesday, Feb 01, 2012 – – – –

– – – – Thursday, Feb 02, 2012 – – – –

– – – – Friday, Feb 03, 2012 – – – –

Set Theory Seminar

Friday, February 3, 2012 10:00 am GC 6417

Mr. Norman Perlmutter (CUNY Graduate Center)

Iterated Prikry Forcing

Abstract. One-step Prikry forcing kills a measurable cardinal by adding a cofinal omega-sequence to it. What happens when we try to iterate this forcing to add cofinal omega-sequences to each measurable cardinal in a set A? If A contains none of its limit points, things work out easily, because at each stage the previous forcing is small. In case A contains some of its limit points, the situation is much more complicated. Magidor developed the technique of iterated Prikry forcing to add cofinal omega-sequences to an arbitrary set of measurable cardinals. Gitik later revised his technique and figured out how to do the forcing with Easton support rather than Magidor’s full support. Gitik also generalized the technique to Prikry-type forcing. Prikry-type forcing is forcing of the form (P, leq, leq*), where leq* is a suborder of leq, and for every condition p and every formula Phi of the forcing language, there exists q leq *p such that q decides Phi. In the usual Prikry forcing, leq* corresponds to extensions where the stem stays the same but the measure one set may be made smaller. I will develop the basics of iterated Prikry forcing, following the exposition from Gitik’s article in the Handbook of Set Theory.

Model Theory Seminar

Friday, February 3, 2012 12:30 pm GC 6417

Mr. Thomas Ferguson (Ph.D. Program in Philosophy, Graduate Center of CUNY) Model Theory for Many-Valued Logics, I: Tutorial and Motivations

Abstract. A number of generalizations of the classical predicate calculus have been made over the years; one quite simple path is to increase the number of possible truth values to which a formula may be mapped. In preparation for discussing the model theory of such many-valued logics, this session will be devoted to a tutorial in such logics as well as a discussion of their philosophical motivations and practical applications.

Logic Workshop

Friday, February 3, 2012 2:00 pm GC 6417

Professor Grigor Sargsyan (Rutgers University)

The Solovay Hierarchy

Abstract. It is a well-known and a fascinating phenomenon that the large cardinal hierarchy forms a consistency strength hierarchy, i.e., large cardinal axioms form a linear hierarchy ordered by their consistency strengths and, which is what makes it all very

fascinating, the consistency of almost any known interesting mathematical theory can be reduced to the consistency of a large cardinal axiom. Lets call this the large cardinal phenomenon. In order to argue that large cardinal hierarchy can indeed be taken to be the backbone of all of mathematics, one at least needs to reverse the above consistency reductions and show that the consistency of many natural and interesting mathematical theories entail the consistency of large cardinal axioms. However, such reversals have only been worked out for a very small initial segment of large cardinal hierarchy. Recently the Solovay hierarchy, which is a descriptive set theoretic hierarchy, has been used by several authors to establish such reversals by using the Solovay hierarchy as an intermediary. These new methods have been used to formulate a general program for establishing such reversals at every level of the consistency strength hierarchy. In this talk, we will outline the evidence that supports the aforementioned phenomenon and also outline the new program mentioned above. Some recent results will be stated and hopefully, some proofs will be given.

Next Week in Logic at CUNY:

– – – – Monday, Feb 6, 2012 – – – –

– – – – Tuesday, Feb 7, 2012 – – – –

– – – – Wednesday, Feb 8, 2012 – – – –

– – – – Thursday, Feb 9, 2012 – – – –

– – – – Friday, Feb 10, 2012 – – – –

Set Theory Seminar

Friday, February 10, 2012 10:00 am GC 6417

Mr. Norman Perlmutter (CUNY Graduate Center)

Using iterated Prikry forcing to construct an inverse limit that is a proper subset of the thread set

Abstract. First, if necessary, I will finish up the development of the basics of iterated Prikry forcing from last week’s seminar. Next, I will review material on inverse limits of systems of elementary embeddings of models of set theory, which I first presented in the set theory seminar on October 14, 2011. Finally, I will apply an Easton/Gitik-support Prikry iteration to exhibit a system whose inverse limit exists but is a proper subset of the thread set. In other words, the inverse limit in the category of models of ZFC and elementary embeddings exists but is a proper subset of the inverse limit in the category of sets and set maps. The proof will take advantage of the fact that any tail of a Prikry iteration that does not begin at a limit point of the set of measurables being killed has a dense subset in the ground model. This allows for the iteration to be re-ordered as if it were a product.

Model Theory Seminar

Friday, February 10, 2012 12:30 pm GC 6417

Mr. Thomas Ferguson (Ph.D. Program in Philosophy, Graduate Center of CUNY) Model Theory for Many-Valued Logics, II: Ultraproducts and

Constructing Many-Valued Models

Abstract. In this session, some model theory for many-valued logics will be introduced. It will be shown how techniques such as Los’ Theorem may be extended to such logics and methods of constructing many-valued models from classical models will be introduced. Ultimately, we wish to show that ultraproducts “play nice” with such constructions by obeying some commutative properties.

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