E-Recursion, Forcing and C*-Algebras

The following sections are included:

• CONTENTS
• FOREWORD
• PREFACE

Basics of Elliott's classification program are outlined and juxtaposed with the abstract classification theory from descriptive set theory. Some preliminary estimates on the complexity of the isomorphism relation of separable C*-algebras are given.

In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set theory. For instance using them, one can formulate “forcing axioms” and prove them consistent relative to a supercompact cardinal. Examples are PFA, which says that Martin's axiom holds for proper forcings, and MM, which says the same for semiproper forcings. Both these axioms imply the negation of CH. This is due to the fact that some proper forcings add new reals. Complete forcings, on the other hand, not only add no reals, but also no countable sets of ordinals. Hence they cannot change a cofinality to ω. Thus none of these theories enable us e.g. to show, assuming CH, that Namba forcing can be iterated without adding new reals…

E-recursive versus Σ₁ admissible, two competing intuitions of recursion theory. Partial E-recursive functions, E-recursive enumerability and computation trees. Divergence witnesses and reflecting ordinals. Priority arguments and forcing constructions for E-closed structures. Logic on E-closed structures.