# Sh:513

• Shelah, S. (2002). PCF and infinite free subsets in an algebra. Arch. Math. Logic, 41(4), 321–359.
• Abstract:
We give another proof that for every \lambda\geq\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 §2 we show that large pcf(\mathfrak{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\neq 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 \le\sigma functions there is an infinite independent subset) then for no distinct regular \lambda_i\in {\rm Reg}\backslash\mu^+ (for i<\kappa) does \prod_{i<\kappa}\lambda_i/[\kappa]^{\le\sigma} have true cofinality. We look at IND(\langle J^{bd}_{\kappa_n}: n<\omega\rangle) and more general version, and from assumptions as in §2 get results even for the non stationary ideal. Lastly, we deal with some other measurements of [\lambda]^{\ge \theta} and give an application by a construction of a Boolean Algebra.
• Version 2002-04-03_10 (56p) published version (39p)
Bib entry
@article{Sh:513,
author = {Shelah, Saharon},
title = {{PCF and infinite free subsets in an algebra}},
journal = {Arch. Math. Logic},
fjournal = {Archive for Mathematical Logic},
volume = {41},
number = {4},
year = {2002},
pages = {321--359},
issn = {0933-5846},
mrnumber = {1906504},
mrclass = {03E04},
doi = {10.1007/s001530100101},
note = {\href{https://arxiv.org/abs/math/9807177}{arXiv: math/9807177}},
arxiv_number = {math/9807177}
}