Weak diamond and pcf theory

Abstract

We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from ${\aleph }_{\delta }$ for which $2^{{\aleph }_{\delta }}<{\aleph }_{(|\delta {|}^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.