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Advances in cardinal arithmetic. (English) Zbl 0844.03028

Sauer, N. W. (ed.) et al., Finite and infinite combinatorics in sets and logic. Proceedings of the NATO Advanced Study Institute, Banff, Canada, April 21-May 4, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 355-383 (1993).
Summary: If \(\text{cf}\kappa= \kappa\), \(\kappa^+< \text{cf}\lambda= \lambda\) then there is a stationary subset \(S\) of \(\{\delta< \lambda: \text{cf}(\delta)= \kappa\}\) in \(I[\lambda]\). Moreover, we can find \(\overline C= \langle C_\delta: \delta\in S\rangle\), \(C_\delta\) a club of \(\lambda\), \(\text{otp}(C_\delta)= \kappa\), guessing clubs and for each \(\alpha< \lambda\) we have: \(\{C_\delta\cap \alpha: \alpha\in \text{nacc}C_\delta\}\) has cardinality \(< \lambda\).
We prove that, for example, there is a stationary subset of \({\mathcal S}_{< \aleph_1}(\lambda)\) of cardinality \(\text{cf}({\mathcal S}_{< \aleph_1}(\lambda), \subseteq)\).
We prove the existence of nice filters, where instead of being normal filters on \(\omega_1\) they are normal filters with larger domains, which can increase during a play. They can help us transfer the situation on \(\aleph_1\)-complete filters to normal ones.
We consider ranks and niceness of normal filters, such that we can pass, say, from \(\text{pp}_{\Gamma(\aleph_1)}(\mu)\) (where \(\text{cf}\mu= \aleph_1\)) to \(\text{pp}_{\text{normal}}(\mu)\).
We consider some weakenings of GCH and their consequences. Most have not been proved independent of ZFC.
For the entire collection see [Zbl 0780.00039].

MSC:

03E10 Ordinal and cardinal numbers
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