This Week in Logic at CUNY:

– – – – Monday, Nov 7, 2011 – – – –

– – – – Tuesday, Nov 8, 2011 – – – –

Computational Logic Seminar

Room 3309. Time 2:00 – 4:00 PM

Tuesday, November 8.

Speaker: Andre Rodin

Title: Doing and Showing

Abstract: The persisting gap between the formal and the informal mathematics is due

to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.

– – – – Wednesday, Nov 09, 2011 – – – –

– – – – Thursday, Nov 10, 2011 – – – –

– – – – Friday, Nov 11, 2011 – – – –

Set Theory Seminar

Friday, November 11, 2011 10:00 am GC 6417

Professor Thomas Johnstone (NYC College of Technology (CUNY)) What is ZFC set theory when the power set axiom is removed?

Abstract. When prompted, many set theorists offer the following list of axioms: extensionality, pairing, union, infinity, separation, foundation, replacement and choice. In this talk we will prove that this formulation of set theory without power set is weaker than commonly supposed, and it is inadequate to prove several basic facts often desired in its context. For example, infinite successor cardinals can be singular, Los’ ultrapower theorem can fail, Gaifman’s theorem can fail (i.e. cofinal Sigma_1-elementary embeddings need not be fully elementary), and Sigma_1-formulas need not be closed under bounded quantification. Nevertheless, these deficits are completely repaired if one uses collection, rather replacement in the

axiomatization above. This is joint work between Joel Hamkins, Victoria Gitman and myself, and a pre-print is available at

http://arxiv.org/abs/1110.2430.

Model Theory Seminar

Friday, November 11, 2011 12:30 pm GC 6417

Professor Alfred Dolich (Kingsborough Community College)

Integer parts of real closed fields

Logic Workshop

Friday, November 11, 2011 2:00 pm GC 6417

Professor Samson Abramsky (Oxford University)

Independence in Quantum Foundations and Social Choice

Logic and Games Seminar

Friday, November 11, 2011 4:15 pm GC, Room 4419

Professor Kit Fine (New York University)

Truthmaker Semantics for Intuitionistic Logic

Abstract. tba

Next Week in Logic at CUNY:

– – – – Monday, Nov 14, 2011 – – – –

– – – – Tuesday, Nov 15, 2011 – – – –

– – – – Wednesday, Nov 16, 2011 – – – –

– – – – Thursday, Nov 17, 2011 – – – –

– – – – Friday, Nov 18, 2011 – – – –

The New York Colloquium on Algorithms and Complexity

Friday, November 18, 2011, The Graduate Center, CUNY, Room 4102, 9:00 am to 5:30 pm

NYCAC, the New York Colloquium on Algorithms and Complexity is an annual event. Its purpose is to give the opportunity to graduate students in New York to observe talks of researchers from all areas of the theory of Algorithms and Computational Complexity. Invited Speakers include Eric Allender (Rutgers U.), Martin Fürer (Penn State U.), William Gasarch (U. Maryland), George Karakostas (McMaster U.), Michael Lampis (KTH), Aris Tentes (NYU), Christos Tzamos (MIT), Vassilis Zikas (U. Maryland)

More information, including a full schedule, is available at http://www.sci.brooklyn.cuny.edu/~zachos/nycac4/

Logic Workshop

Friday, November 18, 2011 10:00 am* GC 6417

* NOTE special time

Professor Peter Koepke (Rheinische Friedrich-Wilhelms-Universität Bonn) Violating the Singular Cardinals Hypothesis without Large Cardinals

Abstract. After Easton had proved that the behavior of the exponential function 2κ at regular cardinals κ is independent of the axioms of set theory except for some simple classical laws, attention turned to the situation at singular cardinals. The Singular Cardinals Hypothesis SCH implies that the Generalized Continuum Hypothesis GCH 2κ = κ+ holds at a singular cardinal κ if GCH holds below κ. The SCH has triggered many developments in large cardinals, forcing and inner model theory. Gitik and Mitchell have determined the consistency strength of the negation of SCH in Zermelo Fraenkel set theory with the axiom of choice in terms of measurable cardinals of high Mitchell orders.

Over several years, Arthur Apter and I have pursued a program of determining such consistency strengths in Zermelo Fraenkel set theory without the axiom of choice. Often the consistency strengths become much lower than in the choiceful setting. In my talk I want to present joint work with Moti Gitik that the following is relatively consistent with Zermelo Fraenkel set theory: GCH holds below the first

uncountable limit cardinal ℵω and there is a surjection from its power set P(ℵω) onto some arbitrarily high cardinal λ. The proof uses forcing and symmetric submodels to adjoin λ subsets of ℵω that have a lot of pairwise agreement so that the GCH below ℵω is preserved.

The result lead to the conjecture that without the axiom of choice and without assuming large cardinal strength an – appropriately modified – exponential function can take rather arbitrary values at all infinite cardinals.

Note the special time! This talk begins at 10 am, and is joint between the CUNY Logic Workshop and the Set Theory Seminar. There will be a second Logic Workshop talk today at 2 pm.

Model Theory Seminar

Friday, November 18, 2011 12:30 pm GC 6417

Mr. Manuel Alves (Ph.D. Program in Mathematics, Graduate Center of CUNY) Zilber’s trichotomy

Logic Workshop

Friday, November 18, 2011 2:00 pm GC 6417

Dr. Sam Sanders (Ghent University)

Reuniting the antipodes: bringing together Nonstandard Analysis and Constructive Analysis

Abstract. Constructive Analysis was introduced by Erret Bishop to identify the `computational meaning’ of mathematics. In the spirit of intuitionistic mathematics, notions like `algorithm,’ `explicit computation,’ and `finite procedure’ are central. The exact meaning of these vague terms was left open, to ensure the compatibility of Constructive Analysis with several traditions in mathematics. Constructive Reverse Mathematics (CRM) is a spin-off of Harvey Friedman’s famous Reverse Mathematics program, based on Constructive Analysis. Bishop famously derided Nonstandard Analysis for its lack of computational meaning. In this talk, we introduce `Ω-invariance’: a simple and elegant definition of `finite procedure’ in (classical) Nonstandard Analysis. Using an intuitive interpretation, we obtain many results from CRM, thus showing that Ω-invariance is quite close to Bishop’s notion of `finite procedure.’

This is the second Logic Workshop talk of the day. The first, by Peter Koepke, begins at 10:00 am.

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