Wednesday, 25 February 2009

An assumption in pcf theory

Often in theorems of pcf theory one has the assumption that the length of sequences of functions has to respect some bound so things can be well-defined. For instance, let a=[aleph2,...,alephn,...:n<omega] be a set of regular cardinals, say you have a sequence fbeta in proda of length at most |a|+. Then supbetafbetainproda since |a|+<min(a). But why is this true? If you have for example an omega2 sequence of functions f:kapparightarrowkappa such that f(kappa)inkappa, kappa some alephn, n not 0 and not 1,then why is fbeta for beta=omega2 outside of the product, as far as we know, we don't know if 2aleph0=aleph2 since a is a countable set of regular cardinals (say the set of alephn's)? Thanks

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