Friday, 6 March 2015
14:00 - 14:30 Check in
14:30 - 15:15 C. T. Chong (National Univ. of Singapore)
Title : The coding Lemma

#### Abstract

The coding Lemma refers to the statement that in a model of P^-+ B\Sigma_n, any set that is \Delta_n on a bounded set is coded. This Lemma is the definable version of saturation in a model. We discuss an equivalent statement of the coding Lemma, and present some of its applications in reverse mathematics, including the first order strength of Ramsey's theorem for pairs.
Coffee Break
15:45 - 16:30 Emanuele Frittaion (Tohoku University)
Title : Combinatorial principles via a partition theorem for pairs of rational numbers

#### Abstract

I will discuss the reverse mathematics and computability theory of the following extension of RT22 (Ramsey's theorem for pairs of natural numbers and two colors):

Theorem [Erd\"{o}s, Rado 1952] The partition relation Q \to (\aleph_0, Q)^2 holds.

The theorem says that that for every 2-coloring f: [Q]^2\to 2 of pairs of rational numbers there exists either an infinite 0-homogeneous set or a dense 1-homogeneous set. More explicitly, there exists an infinite set A \subseteq Q such that either f\restriction [A]^2=0 or (A, \leq_Q) is dense and f\restriction [A]^2=1.

I will also talk about analogs of CAC (Chain Anti-Chain) and ADS (Ascending Descending Sequence) related to Q \to (\aleph_0, Q)^2.

16:45 - 17:30 Hiroshi Sakai (Kobe University)
Title : On monadic second order theory of $\omega_2$

#### Abstract

The monadic second order logic is a restriction of the second order logic in which only quantifications over unary predicates are allowed. In many cases, the mondadic second order theories are simple while retaining some extent of expressivity. B\"{u}chi proved that the modadic second order theories of $( \omega , < )$ and $( \omega_1 , < )$ are both decidable. On the other hand, it was proved by Grevich-Magidor-Shelah that the decidability of the monadic second order theory of $( \omega_2 , < )$ is independent of ZFC. In this talk we discuss the decidability of the monadic second order theory of $( \omega_2 , < )$ under natural extensions of ZFC such as ZFC + Forcing Axiom.