### 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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