Ernest Schimmerling

Mathematical logic seminar - April 29, 2004

Time: 1:30 - 3 p.m.

Room: PPB 300

Speaker:     Paul Raff
Undergraduate Honors Student
Department of Mathematical Sciences
Carnegie Mellon University

Title: Cardinal arithmetic : early results of Shelah towards PCF theory

Abstract: In this talk, I will survey a family of results on cardinal exponentiation. All of these theorems concern what is provable in ZFC. The possible behaviors of the continuum function $\kappa \mapsto 2^\kappa$ restricted to regular cardinals was mostly understood in 1970. Attention then turned to singular cardinals. In 1974, Silver showed that if $\lambda$ is a singular cardinal of uncountable cofinality and $2^\kappa = \kappa^+$ for all $\kappa < \lambda$, then $2^\lambda = \lambda^+$. Silver's proof was innovative in that it used generic ultrapowers. Later, direct combinatorial proofs of Silver's theorem were found and used to find more explicit bounds. For example, Galvin and Hajnal showed in 1975 that if $\aleph_\alpha$ is a singular strong limit cardinal of uncountable cofinality, then $2^{\aleph_\alpha} < \aleph_{(2^{|\alpha|})^+}$.

The theorems above are all about powers of cardinals of uncountable cofinality. By the late 1980's, Shelah had developed techniques, known collectively as {\em PCF theory}, which could also be used to prove theorems about cardinals of countable cofinality. Perhaps most famously, Shelah showed that if $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega} < \aleph_{\omega_4}.

This talk is mainly concerned with a transitional paper by Shelah called "On Power of Singular Cardinals" [Shelah 111], which deals only with uncountable cofinalities. In this paper, Shelah gives a proof of the Galvin-Hajnal theorem that uses generic ultrapowers much the same way Silver did. To make it work, Shelah introduces game theoretic techniques for guaranteeing that the generic ultrapowers are sufficiently wellfounded. Shelah uses these methods to obtain new results of the form: if $\aleph_\alpha$ is a singular strong limit cardinal of uncountable cofinality and $\aleph_\alpha$ is small, then $(2^(2^{\aleph_\alpha})^+$ is also small. There are various meanings of {\em small}, for example, {\em less than the least weakly inaccessible}.