PCF and infinite free subsets in an algebra

by Shelah. [Sh:513]
Archive for Math Logic, 2002
We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega . In section 2 we show that large pcf (a) implies existence of free sets. An example is that if pp (aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l< omega}) . Then we present results complementary to those of section 2 (but not close enough): if IND (mu, sigma) (in every algebra with universe lambda and <= sigma functions there is an infinite independent subset) then for no distinct regular lambda_i in Reg backslash mu^+ (for i< kappa) does prod_{i< kappa} lambda_i/[kappa]^{<= sigma} have true cofinality. We look at IND (< J^{bd}_{kappa_n}: n< omega >) and more general version, and from assumptions as in section 2 get results even for the non stationary ideal. Lastly, we deal with some other measurements of [lambda]^{>= theta} and give an application by a construction of a Boolean Algebra.

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