The power of aleph-omega

The formula aleph_omega^(aleph_0) <= 2^(aleph_0)+ 
aleph_(omega_4) is only one of the consequences of pcf theory. While Jack Silver proved that GCH below a singular cardinal of uncountable cofinality implies GCH AT that cardinal, Menachem Magidor proved that GCH below aleph_omega does not prove GCH at aleph_omega.

pcf theory, as developed in Shelah's book "Cardinal Arithmetic", was developed not only to compute or estimate the values of the gimel function kappa^(cf(kappa) [from which one can then compute arbitrary powers kappa^lambda], but rather to analyse the order-theoretic structure of products kappa^(cf(kappa)) for singluar cardinals kappa, or more generally , of products of the form product_i lambda_i , where (lambda_i: i < delta) is a short sequence of regular cardinals.

The table of contents for the "Cardinal arithmetic" book, together with the introduction, is available as a DVI file.

Saharon Shelah has also written several other Books.