This Week in Logic at CUNY:

– – – – Monday, May 7, 2012 – – – –

– – – – Tuesday, May 8, 2012 – – – –

Computational Logic Seminar

Time 2:00 – 4:00 PM, May 8, Room 3209.

Speaker: Sergei Artemov (Graduate Center)

Title: Joining provability and explicit proofs.

Abstract: We will study the system GLA that combines provability logic

and the logic of proofs. The principal results of this talk were

obtained by E. Nogina and we will follow her work closely.

– – – – Wednesday, May 09, 2012 – – – –

Models of Peano Arithmetic

Wednesday, May 9, 2012 6:30 pm Room 4214-03

Professor Leszek Kolodziejczyk (Institute of Mathematics, University

of Warsaw)

Open Induction-style methods in the study of fragments of bounded arithmetic

Abstract. Sam Buss’ bounded arithmetic S_2, defined in the mid-80’s,

has been studied mostly because of its connections with computational

complexity and propositional proof complexity. The main goal in the

area, to separate natural fragments of S_2 from the full theory, is

apparently out of reach except for some very weak fragments. For this

reason, attention has largely shifted to somewhat different questions,

but a better understanding of the border between “very weak”

subtheories of S_2 and those “too strong for current methods” is

still desirable. In this talk, I will discuss some separations

obtained by techniques inspired by open induction, first introduced

into bounded arithmetic by Boughattas and Ressayre.

– – – – Thursday, May 10, 2012 – – – –

– – – – Friday, May 11, 2012 – – – –

Model Theory Seminar

Friday, May 11, 2012 12:30 pm GC 6417

Professor Philipp Rothmaler (The City University of New York, BCC)

More examples: definable subgroups of modules and Ziegler spectra of rings

Abstract. This talk is first of all intended as illustration to the

previous talk on modules in that I will present more examples of pp

definable subgroups and more pictures. I will then introduce the

Ziegler spectrum of a ring, a central concept of the model theory of

modules, and compute it for the integers. Some other examples of

Ziegler spectra will be mentioned as well.

Logic Workshop

Friday, May 11, 2012 2:00 pm GC 6417

Professor Philipp Rothmaler (The City University of New York, BCC)

Definable subcategories and Mittag-Leffler modules

Abstract. Given an associative ring R, a definable subcategory is a

subcategory of R-Mod, the category of all left R-modules, that is

closed under pure submodule, direct sum (or product), and direct limit

(=colimit). It can be shown that these are indeed definable, namely,

they are exactly the subcategories axiomatized by pp implications

(i.e. by axioms of the form F–>G, where F and G are (1-place) pp

formulas), and they are in bijective correspondence with the closed

subsets of the Ziegler spectrum (cf. preceding talk in Model Theory

Seminar). Following Raynaud and Gruson (1971), a left R-module M is

called Mittag-Leffler if the canonical map (\prod K_i) \tensor M –>

\prod (K_i \tensor M) is injective for all collections of K_i from

K=Mod-R. In old work (1994), I extended this to arbitrary subclasses K

of Mod-R in the obvious way and proved that M is K-Mittag-Leffler iff

M is, in the model-theoretic sense, positively atomic with respect to

the dual class DK of K in R-Mod. Here D is Prest’s elementary duality

(as introduced in Alves’ earlier talk, cf. today’s talk in the Model

Theory Seminar). As a consequence one sees that it suffices to

consider definable subcategories K (and DK) in the above. An

interesting special case first investigated by Goodearl (1972) is when

K consists of the single module R (considered as a right R-module) or,

equivalently, when K consists of all flat right R-modules. Then DK is

the class of all absolutely pure left R-modules. It is interesting to

note that these classes are different from the definable categories

they generate precisely when the ring is not left coherent.

I will explain all of the above and prove some recent structure results.

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