This Week in Logic at CUNY

This Week in Logic at CUNY:
– – – – Monday, Sep 19, 2011 – – – –

– – – – Tuesday, Sep 20, 2011 – – – –

Computational Logic Seminar
Room 3309, 2:00 – 4:00 PM, September 20, 2011
Speaker: Melvin Fitting (Graduate Center CUNY)
Title: Possible World Semantics for FOLP.

Abstract. First Order Logic of Proofs (FOLP) has been formulated by Artemov and Yavorskaya. Their work includes an axiomatic formulation, an arithmetic completeness theorem, and a realization theorem. Here we add to all this a possible world semantics, and an Mkrtychev semantics, for FOLP, with appropriate soundness and completeness proofs. We also show a first order analog of completeness with respect to fully explanatory models. We believe the possible world semantics provides additional insight into the handling of the quantifiers in FOLP. It is remarkable that the first order results do not involve any special tricks, but instead are natural extensions of what worked in the propositional setting.

– – – – Wednesday, Sep 21, 2011 – – – –

– – – – Thursday, Sep 22, 2011 – – – –

– – – – Friday, Sep 23, 2011 – – – –

Set Theory Seminar
Friday, September 23, 2011 10:00 am GC 6417
Dr. David Schrittesser
The ineffable list property for omega_2 is consistent with 2^omega>omega_2

Abstract. The ineffable list property, or what Weiss calls the “combinatorial essence of supercompatcness” is a combinatorial principle generalizing the tree property. It can be thought of as “supercompactness minus inaccessibility”, in that it is equivalent to supercompactness for inaccessible cardinals, but can itself hold for small cardinals. Under PFA, omega_2 has the thin list property, i.e. ITP(omega_2) holds — and Weiss and Viale have recently used this to prove a landmark result about the consistency strength of PFA. A different question is which consequences of PFA might already be implied by ITP(omega_2). Maybe your intuition is that this holds for none of the interesting consequences? In joint work with Shelah, we confirm this intuition in the case of the size of the continuum.

Model Theory Seminar
Friday, September 23, 2011 12:30 pm GC 6417
Professor Roman Kossak (The City University of New York)
Models and types of PA, III

Abstract. I will survey basic results concerning cofinal extensions of models of PA.

Logic Workshop
Friday, September 23, 2011 2:00 pm GC 6417
Professor Gunter Fuchs (The City University of New York)
Set-theoretic geology of canonical inner models

Abstract. In joint work with Hamkins and Reitz, we introduced inner models that are obtained by reversing the method of forcing, in the sense that we look down at the ground models of a given universe. Taking the intersection of these ground models results in what we call the mantle, and taking the intersection of all ground models of all set forcing extension yields the generic mantle. One of our main results on these models (the mantle and the generic mantle) is that they are not canonical — in fact, any model of set theory can be extended by class forcing to a model of set theory whose mantle and generic mantle is precisely the original model. In particular, the mantle of a model can itself be a nontrivial forcing extension of another model. There are many open questions around set-theoretic geology: we don’t know whether the mantle can ever be different from the generic mantle, and we don’t know whether the generic mantle always is a model of ZFC. One of the major basic questions that is relevant in this context is whether the ground models of a given model always are downward directed. In general, we don’t have an answer, but there are partial results. For example, if the universe is
constructible from a set, then its grounds are downward directed (in a strong sense). This motivates the investigation of the set-theoretic geology of canonical inner models, which was begun by Ralf Schindler and myself. I am going to give an overview of some initial results.

Logic and Games Seminar
Friday, September 23, 2011 4:15 pm GC, Room 4419
Professor Olympia Hadjiiliadis (Brooklyn College and Graduate Center, The City University of New York)
Preventing market crashes through insuring the speed of drawdowns

Abstract. In this work we derive analytical formulas for the joint distribution of the drawdown, the last visit time of the maximum of a process preceding the drawdown and the maximum of the process under general diffusion dynamics. The motivation for this work arises in the financial risk management of drawdowns. Drawdowns measure the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level. Therefore a quantity related to the drawdown is the duration of time between the drawdown and the last time at which the maximum was achieved. This quantity is studied here.

Next Week in Logic at CUNY:
– – – – Monday, Sep 26, 2011 – – – –

– – – – Tuesday, Sep 27, 2011 – – – –

– – – – Wednesday, Sep 28, 2011 – – – –

– – – – Thursday, Sep 29, 2011 – – – –

– – – – Friday, Sep 30, 2011 – – – –

Set Theory Seminar
** NOTE: special time this week **
Friday, September 30, 2011 12:00 pm GC 6417
Professor Katie Thompson (Carnegie Mellon Univeristy)
Mitchell-order configurations of normal ultrafilters at measurable cardinals

Abstract. Given a measurable cardinal kappa, let U and U’ be two normal ultrafilters on kappa. We say that U is Mitchell-below U’ iff U is an element of the ultrapower V^kappa / U’. This gives a
well-founded partial ordering on all such ultrafilters on kappa. There are several known models in which specific Mitchell-order
configurations can be observed, but it is still unknown whether certain patterns are consistent with ZFC, in particular the 4 ultrafilter configuration in the shape of an ‘N’. Friedman and Magidor recently answered some open questions regarding the number of normal ultrafilters at a measurable cardinal by using the so-called ‘tuning fork method’ and coding techniques. James Cummings and I have been investigating these techniques to not only control the number of ultrafilters, but also their Mitchell-order. We have some partial results but this is work in progress.

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