This Week in Logic at CUNY

This Week in Logic at CUNY:

– – – – Monday, Oct 3, 2011 – – – –

– – – – Tuesday, Oct 4, 2011 – – – –

** CUNY follows Friday Schedule **
NOTE: there is no Computational Logic Seminar today

– – – – Wednesday, Oct 05, 2011 – – – –

– – – – Thursday, Oct 06, 2011 – – – –

– – – – Friday, Oct 07, 2011 – – – –

** CUNY holiday **

Next Week in Logic at CUNY:

– – – – Monday, Oct 10, 2011 – – – –

– – – – Tuesday, Oct 11, 2011 – – – –

Computational Logic Seminar
Room 3309, 2:00 – 4:00 PM
October 11, 2011
Speaker: Ren-June Wang (Graduate Center) – dissertation proposal Title: Timed Modal Epistemic Logic

– – – – Wednesday, Oct 12, 2011 – – – –

– – – – Thursday, Oct 13, 2011 – – – –

– – – – Friday, Oct 14, 2011 – – – –

Model Theory Seminar
Friday, October 14, 2011 12:30 pm GC 6417
Dr. Mauro Di Nasso (University of Pisa)
Nonstandard analysis in combinatorics of numbers: some examples

Abstract. By using nonstandard analysis, R. Jin [3] showed a beautiful property of sumsets of integers, namely that A + B = {a + b | a ∈ A; b ∈ B} is piecewise syndetic whenever A and B have positive Banach density. (A set of integers is piecewise syndetic if it has bounded gaps on arbitrarily large intervals. The Banach density is a refinement of the upper asymptotic density.) That result raised the attention of several researchers, and has been re-proved by measure-theoretic techniques by V. Bergelson, H. Furstenberg and B. Weiss [2], obtaining an improvement. Last year, M.Beiglboeck [1] found a really nice ultrafilter proof of Jin’s theorem.

Starting from his initial work, R. Jin applied again nonstandard methods to attack density problems in additive number theory, producing a series of relevant results (see e.g. [4,5]). Interesting applications of nonstandard methods have also been recently obtained by S. Leth, which are related to the plane fixed point problem (see [6]). The goal of my talk is to present three examples of new proofs in combinatorial number theory that suggest nonstandard analysis as a useful tool in this area of research.

The first example is about intersection properties of sets of differences A − A which only depend on relative density. The second example is an ultrafilter proof of the partition regularity of injective solutions of linear diophantine equations (the point here is that, in a nonstandard setting, ultrafilters can be identified with points, namely hypernatural numbers). This technique also showed potentiality in the study of some non-linear equations. The third example is a property that improves on Jin’s theorem, which is obtained by elementary combinatorial arguments applied in the nonstandard setting.

(Bibliography appears on web page:
http://nylogic.org/ModelTheory/Fall2011/MauroDiNasso)

Logic Workshop
Friday, October 14, 2011 2:00 pm GC 6417
Professor Philip Ehrlich (Dept. of Philosophy, Ohio University) The absolute arithmetic continuum

Abstract. In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including -ω, ω-1, and a square root of ω, to name only a few, where (as the notation suggests) ω is the least infinite ordinal. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers -— construed here as members of ordered “number” fields —- be individually definable in terms of sets of NBG (von Neumann-Bernays-Gödel set theory with global choice), it may be said to contain “all numbers great and small.” In this respect, No bears much the same relation to ordered fields that the system R of real numbers bears to Archimedean ordered fields.

We suggest that whereas R should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG). We draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.

In addition to its inclusive structure as an ordered field, No has a rich algebraico-tree-theoretic structure that emerges from the recursive clauses in terms of which it is defined. In our treatment of No, which differs markedly from Conway’s treatment, the
algebraico-tree-theoretic features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing
characterizations of No as an absolute continuum.

Seminar in Logic and Games
Friday, October 14, 2011
CUNY Graduate Center, room 4419
4:15 PM
Yair Tauman (Economics – Stony Brook)

– – – – Other Logic News – – – –

– – – – Web Site – – – –

The majority of this information, including an interactive calendar of future events, can be found at our website:

http://nylogic.org/Calendar

Leave a Reply

Your email address will not be published. Required fields are marked *

Time limit is exhausted. Please reload CAPTCHA.