W Hugh Woodin

Definability in the enumeration degrees

Abstract. We prove that every countable relation on the enumeration degrees,

E-recursion, forcing and C*-algebras

{\frak E}

Σ1-Collection and the finite injury priority method

, is uniformly definable from parameters in

Chapter 4: Consequences in Measure Theory

{\frak E}

Chapter 1: Baire Category Theorem and the Baire Category Numbers

. Consequently, the first order theory of

Chapter 15: Five Linear Orderings

{\frak E}

Chapter 11: Biorthogonal Systems

is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order theory of the enumeration degrees of the

Chapter 17: Reflection Principles

\Sigma^0_2

Chapter 14: Five Cofinal Types

sets is not decidable.

Chapter 10: Applications to the S-space Problem and the von Neumann Problem

This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers, and mathematicians.