Asian Initiative for Infinity (AII) Graduate Summer School
(28 Jun - 23 Jul 2010)

Jointly funded by the John Templeton Foundation

~ Abstracts ~


Prikry-type forcings and short extenders forcings
Moti Gitik, Tel Aviv University, Israel

We plan to cover the following topics:
Basic Prikry forcing, tree Prikry forcing, supercompact Prikry forcing, negation of the Singular Cardinal Hypothesis via blowing up the power of a singular cardinal, Extender Based Prikry forcing, forcings with short extenders -gap 2, gap 3, arbitrary gap, dropping cofinalities, some further directions.

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Reverse mathematics of combinatorial principles
Denis Hirschfeldt, The University of Chicago, USA

Computability theory and reverse mathematics provide tools to analyze the relative strength of mathematical theorems. This analysis often reveals surprising relationships between results in different areas, such as the tight connection between nonstandard models of arithmetic, the compactness of Cantor space, and results as seemingly diverse as the existence of prime ideals of countable commutative rings, Brouwer's fixed point theorem, the separable Hahn-Banach Theorem, and Gödel's completeness theorem, among many others. It also allows us to give mathematically precise versions of statements such as "Adding hypothesis A makes Theorem B strictly weaker", or "Technique X is essential to proving Theorem Y".

Combinatorial principles, such as versions of Ramsey's Theorem or results about partial and linear orders, are a particularly rich source of examples in computable mathematics and reverse mathematics. This course will focus on fundamental techniques and themes in this area, with the goal of preparing students to tackle open problems, several of which will be discussed during the course.

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The theory of possible cofinalities (PCF) and some applications
Menachem Magidor, Hebrew University of Jerusalem, Israel

The theory of Possible Cofinalities is the theory developed by Shelah which uncovers the deeper structure below cardinal arithmetic. The main concept is (for a set of regular cardinals A) the set of possible cofinalities of A (pcf(A)) which is the set of the regular cardinals that can be realized as the cofinality of some ultraproduct of A. It turned out that there are many deep results about this operation (as well as fascinating problems).

The Theory has many applications. This course will develop the basic concepts of the theory, will prove the main results like the bound on and (time permitting) will give some other applications like the existence of Jonson Algebras, the impossibility of certain cases of Chang's Conjecture and more.

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Towards PCF
Omer Zilberboim, Hebrew University, Israel

We will define the notions of closed unbounded sets, stationary sets and club guessing sequences, and theorems on them which will be used in Prof. Magidor's lectures.

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