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Canonical Ramsey Theorem with a largenesscondition

A. Khamseh (1)

(1) Department of Mathematical Sciences, Isfahan University of Technology,84156-83111 Isfahan, Iran

For a function f with domain [X]n, where X ⊆ N, we say that H ⊆ Xis canonical for f if there is a υ ⊆ n such that for any x0, . . . , xn−1 andy0, . . . , yn−1 in H, f(x0, . . . , xn−1) = f(y0, . . . , yn−1) iff xi = yi for all i ∈ υ.Canonical Ramsey Theorem is the statement that for any n ∈ N, if f : [N]n →N, then there is an infinite H ⊆ N canonical for f . In this talk we study thefinite version of the Canonical Ramsey Theorem with a largeness condition.

Some of recent trends in stability and NIP theories

Alireza Mofidi

IPM

Abstract

In his program for the classification of first order theories, S. Shelah introduced anddeveloped a machinery called stability theory. Also he introduced several dividing linesin the class of first order theories on the base of the particular combinatorial complexitiesof the theories. Stable theories, NIP theories and simple theories are among the mostimportant of these classes. Later on, several people extended the theory and applied itto the different parts of mathematics. Currently this topic is an active part of the modeltheory and developing techniques in some of the mentioned classes such as NIP classis highly under consideration. In this lecture we talk about some of the main notions,theorems and techniques on which the machinery of stability and its newer versions(sometimes called the neo-stability theory) are based.

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Decidability and Undefinability: A Case for Quantifier Elimination

Saeed Salehi

University of Tabriz, and IPM

E-mail: saeedsalehi@ipm.ir URL: http://saeedsalehi.ir/

Providing a recursively enumerable set of axioms for a mathematical structure is equivalent to givingan algorithm for deciding the theory of that structure: given a first-order formula ϕ of the language L,either ϕ or ¬ϕ is true in the structure M = (M,L), so exactly one of them belongs to the consequencesof the axioms, which is a recursively enumerable set. Thus, the theory of M is r.e. and co-r.e., hencedecidable; or in the other words, one can decide for a give formula ϕ whether M |= ϕ or not.

A well-known theorem of Cantor states that any two countable dense linear orders without endpointsare isomorphic. Thus (Q, <) is the only countable model for the theory of “dense linear orders withoutendpoints”, and so this theory is complete. Whence, the first-order theories of (Q, <) and (R, <) areidentical, though we know that these two structures are completely different (the latter is complete inthe sense that any subset which is bounded above has a supremum, while the former is not).

Putting another way, one can say that the theory of “dense linear orders without endpoints” com-pletely axiomatizes the first-order theories of the structures (Q, <) and (R, <), and so these structuresare decidable. Another way of proving this fact is by “Quantifier Elimination” which is a helpful tool forestablishing decidability of mathematical structures, and also for showing the undefinability of certainsets in structures. One can also show that the theory of “discrete orders without endpoints” completelyaxiomatizes the theory of (Z, <), and the theory of (N, <) can be axiomatized as “discrete order witha least point and no last point”.

This settles the theory of order in the sets N, Z, Q and R. In this talk we will study the theories ofthe following structures:

N Z Q R C{<} 〈N, <〉 〈Z, <〉 〈Q, <〉 〈R, <〉 –{+} 〈N,+〉 〈Z,+〉 〈Q,+〉 〈R,+〉 〈C,+〉{·} 〈N, ·〉 〈Z, ·〉 〈Q, ·〉 〈R, ·〉 〈C, ·〉{+, <} 〈N,+, <〉 〈Z,+, <〉 〈Q,+, <〉 〈R,+, <〉 –{+, ·} 〈N,+, ·〉 〈Z,+, ·〉 〈Q,+, ·〉 〈R,+, ·〉 〈C,+, ·〉{·, <} 〈N, ·, <〉 〈Z, ·, <〉 〈Q, ·, <〉 〈R, ·, <〉 –

{+, ·, <} 〈N,+, ·, <〉 〈Z,+, ·, <〉 〈Q,+, ·, <〉 〈R,+, ·, <〉 –

Surprisingly, we will see that some theories in this table are missing in the literature; i.e., have notbeen studied before. One example is the theory (Q, ·) which is decidable, but no proof of it can befound. The theories (R, ·) and (C, ·) are also decidable, because by a theorem of Tarski, the structures(R,+, ·) and (C,+, ·) are decidable. We give a new proof for the decidability of (R, ·) and (C, ·) withoutappealing to Tarski’s result. Also a new proof for the decidability of the structure (R, ·, <) can be givenwithout using Tarski’s theorem. We also show that the theory (Z, ·, <) is undecidable, and the theory(Q, ·, <) is decidable. These are new theorems with novel and nontrivial proofs. All in all we completethe picture as below, where decidability is indicated by ∆1 and undecidability by ∆1/\ :

N Z Q R C{<} ∆1 ∆1 ∆1 ∆1 –{+} ∆1 ∆1 ∆1 ∆1 ∆1

{·} ∆1 ∆1 ∆1 ∆1 ∆1

{+, <} ∆1 ∆1 ∆1 ∆1 –{+, ·} ∆1/\ ∆1/\ ∆1/\ ∆1 ∆1

{·, <} ∆1/\ ∆1/\ ∆1 ∆1 –

At the end, we will discuss some new results and some open problems when the exponential functionis added into the language.

Computable Analysis and Some Applications

Nazanin Roshandel Tavana

Institute For Fundamental Sciences (IPM), Tehran, Iran

November 5, 2012

Abstract

Computable analysis is a branch of computability theory studying those functions on the real

numbers and related sets which can be computed by machines such as digital computers. The

increasing demand for reliable software in scientific computation and engineering requires a sound

and broad foundation not only of the analytical/numerical but also of the computational aspects of

real number computation. The central subject of this approach of computable analysis is ”Type-2

Theory of Effectivity” (TTE), one of the approaches to effective analysis being discussed today. It

is based on definitions of computable real numbers and functions by A. Turing, A. Grzegorczyk and

D. Lacombe. A framework of concrete computability on finite and infinite sequences of symbols is

introduced. Computability on finite and infinite sequences of symbols can be transferred to other

sets by using them as names. First, computability induced by naming systems is discussed. Then,

computable real numbers and functions are introduced. Also, computability on some special spaces

as metric spaces is discussed. Afterward, Some applications of this method to study effectivity of

metric model theory and measure theory are presented.

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