Ernest Schimmerling

Mathematical logic seminar - October 14, 2004

Time: 12 - 1:20 p.m.

Room: Baker Hall 150

Speaker:     Andreas Liu
Department of Mathematics
Hebrew University of Jerusalem

Title:     Cardinal arithmetic since Silver

Abstract:     A famous theorem of Silver, published in 1974, states that if the generalized continuum hypothesis holds below a singular cardinal of uncountable cofinality, then it holds also at that cardinal. This result surprised almost everyone, and raised hopes that cardinal arithmetic might contain a substantial "core" theory immune to forcing.

Since then, Shelah's pcf theory has justified these hopes to a great extent; along the way, it has changed the language of cardinal arithmetic by focusing attention on relatively concrete objects whose properties are often fixed by the ZFC axioms. In particular, it has proven fruitful to study the "pseudopower" function in place of the classical power function on singular cardinals.

The talk will introduce pseudopowers, survey some results and questions connected with them, and describe how methods from pcf theory can be used to extend Silver's theorem in several directions.