Sh:513

• Shelah, S. (2002). PCF and infinite free subsets in an algebra. Arch. Math. Logic, 41(4), 321–359. arXiv: math/9807177 DOI: 10.1007/s001530100101 MR: 1906504
• 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 $a_n\notin c\ell_M(\{a_l: l\neq n, l<\omega\})$. 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 $\lambda_i\in {\rm Reg}\setminus\mu^+$ (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)

@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}
}