This Week in Logic at CUNY:

– – – – Monday, Oct 17, 2011 – – – –

– – – – Tuesday, Oct 18, 2011 – – – –

Computational Logic Seminar

Room 3309. Time 2:00 – 4:00 PM

Tuesday, October 18.

Speaker: Rohit Parikh (Brooklyn College and CUNY Graduate Center) Title: Knowledge and Uncertainty

Abstract: We consider three issues.

– How does an agent choose in the face of uncertainty? Suppose an agent wants to choose between actions L and R. Each has a number of possible consequences, Some consequences of L are better than some of R and vice versa. Then how to choose?

– We show that there are various versions of rationality depending on whether an agent is cautious or aggressive or moderate. The famous experiment (reported by Kahneman) about the Asian Disease question can be explained as a switch from one version of rationality to another. – Suppose agent M is in a position to control the knowledge that agents A and B (who are playing a game) have. Once their state of knowledge is determined they will play the game in a certain way. How should M pick the knowledge states of A and B so as to maximize the benefit to M herself.

Given the desired knowledge states of A and B (not only of facts but also of each other) we show that M can send signals to A and B so as to achieve this state of knowledge. Indeed a single (but complex) signal to each of A and B will suffice.

Note: This work was performed jointly with Cagil Tasdemir and Andreas Witzel (NYU). It is related to older work by Kannai, Peleg, Fishburn, Tauman, Zamir as well as to more recent work by Artemov and Aumann.

– – – – Wednesday, Oct 19, 2011 – – – –

– – – – Thursday, Oct 20, 2011 – – – –

– – – – Friday, Oct 21, 2011 – – – –

Set Theory Seminar

Friday, October 21, 2011 10:00 am GC 6417

Mr. Jay Williams (Rutgers)

Group embeddability and countable Borel quasi-orders

Abstract. Descriptive set theory gives us a framework for analyzing the relative complexity of quasi-orders (i.e. reflexive transitive relations) arising in many areas of mathematics, such as Turing reducibility of sets of natural numbers or embeddability of countable groups, using the notion of a Borel reduction. I will discuss a special class of quasi-orders, the countable Borel quasi-orders, and focus in particular on embeddability of finitely-generated groups. The ideas will apply to the more general case of embeddability of countable groups.

Model Theory Seminar

Friday, October 21, 2011 12:30 pm GC 6417

Professor Roman Kossak (The City University of New York)

Models and types of PA, IV

Abstract. Countable recursively saturated and short recursively saturated models will be used in an example of an abstract elementary class with a special property.

Logic Workshop

Friday, October 21, 2011 2:00 pm GC 6417

Dr. Daisuke Ikegami (University of Helsinki)

Ω-logic and Boolean-valued 2nd-order logic

Abstract. Ω-logic discusses “forcing absoluteness”, which is the preservation of truth-values of statements between ground models and their set forcing extensions. Ω-valid sentences are those which are true in any set forcing extension, while Ω-provable sentences are those which are true in all transitive models of ZFC that are closed under some “universally Baire sets”, that are the key notion tying forcing absoluteness with large cardinals. The Ω-conjecture states that those two collections of sentences are exactly the same assuming the existence of a proper class of Woodin cardinals, that are the essential large cardinals when discussing forcing absoluteness. Although the Ω-conjecture is consistent and its truth value does not change by forcing, it is still open whether it is true or not.

Boolean-valued 2nd-order logic is a Boolean-valued logic for 2nd-order statements: “subsets” and “relations” of a given 1st-order universe M can be seen as functions sending n-tuples of elements of M to elements of a given complete Boolean algebra B and one can assign an element of B to each 2nd-order formula as a truth value. Boolean valid sentences are those 2nd-order sentences whose truth values are 1 in any such assignment while one can define Boolean provable sentences with the help of universally Baire sets and suitable “Henkin models,” which are the key notion when investigating 2nd-order logic. We do not know whether those two collections of sentences are the same while any Boolean provable sentence is Boolean valid.

In this talk, we introduce Ω-logic and Boolean-valued 2nd-order logic and discuss the connection between them. We show the following two things:

1. Ω-validity is as complex as Boolean-validity. As a corollary, Boolean-validity is Δ2 in set theory assuming Ω-conjecture.

2. If Boolean valid sentences are exactly the same as Boolean provable sentences, then the Ω-conjecture holds.

The first result contrasts the fact that the validity of full 2nd-order logic is Π2-complete in set theory.

This is joint work with Jouko Väänänen.

Seminar in Logic and Games

Friday, October 21, 2011, 4:15 PM

Room 4419, CUNY Graduate Center

Matthew Moore (Professor and Chairperson, Philosophy Department, Brooklyn College)

Mathematical Realism, Abductively

Abstract. The first half of this paper summarizes an argument for mathematical realism, the thesis that the statements of classical mathematics are both truth-valued and true. The argument, which is assembled largely from materials found in the writings of Burgess, Quine and Resnik, proceeds abductively, by offering realism as the best explanation of certain gross feaures of mathematical and scientific practice. The second half of the paper considers what ontology for mathematics best comports with realism thus defended, and settles on a version of structuralism according to which mathematical structures are ante rem universals.

Next Week in Logic at CUNY:

– – – – Monday, Oct 24, 2011 – – – –

– – – – Tuesday, Oct 25, 2011 – – – –

– – – – Wednesday, Oct 26, 2011 – – – –

– – – – Thursday, Oct 27, 2011 – – – –

– – – – Friday, Oct 28, 2011 – – – –

Logic Workshop

Friday, October 28, 2011 2:00 pm GC 6417

Professor Benedikt Löwe (Universiteit van Amsterdam)

Multiplication in the hierarchy of norms

Abstract. A norm is a surjective function from the reals onto an ordinal. Given two norms φ and ψ, we can define a Wadge-like game G(φ ψ): if player I plays x and player II plays y, player II wins if and only if φ(x) ≤ ψ(y). If player II has a winning strategy in G(φ ψ), we write φ ≤ ψ . The equivalence classes of the derived equivalence relation form the hierarchy of norms.

Under the assumption of the axiom of determinacy (AD), the hierarchy of norms is a wellordering. In [1], Duparc investigated the Borel fragment of the hierarchy; in [2], we investigated the full hierarchy under the assumption of AD, and gave a lower bound of Θ2 for its length (where Θ := sup {α : there is a surjection from R onto α}). In this talk, we investigate further closure properties of the hierarchy of norms, in particular, the existence of an analogue of the multiplication for sets in the Wadge hierarchy, i.e., an operation such that (for well-chosen φ and ψ) the ordinal rank of φ⊗ψ is at least as big as the product of the ordinal ranks of φ and ψ. This closure property allows us to improve the lower bound to Θω.

(Bibliography available at http://nylogic.org/Workshop/Fall2011/BenediktLoewe)

Seminar in Logic and Games

Friday, October 28, 2011, 4:15 PM

Room 4419, CUNY Graduate Center

Jan Plaza (Computer Science – SUNY Plattsburgh)

– – – – Other Logic News – – – –

Philosophy of set theory course at NYU

J. D. Hamkins is running a course at NYU on the philosophy of set theory, meeting Tuesdays 11-1 at NYU Philosophy Department, 5 Washington Place. Topics for next week Tuesday include P. Maddy’s articles on “Believing the axioms” and C. Freiling’s article, “Axioms of Symmetry: throwing darts at the real number line”. Future topics will include philosophical writings on reflection, large cardinals, categoricity, and the question of pluralism and indeterminateness in set theory. Please contact J. D. Hamkins (jhamkins@gc.cuny.edu) for further information.

– – – – Web Site – – – –

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