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Basis problems in combinatorial set theory. (English) Zbl 0918.03025

The author discusses results and conjectures concerning some basis problems in set theory. A basis for a class \(S\) of structures is a collection \(S_0\) of critical members of \(S\) such that any given structure from \(S\) can be related via some kind of connecting map to one member from \(S_0\). For instance, the author showed that under PFA, the five sets 1, \(\omega\), \(\omega_1\), \(\omega\times \omega_1\) and \(\text{Fin}_{\omega_1}\) (the collection of all finite subsets of \(\omega_1\) ordered by inclusion) form a basis for the class of all directed sets on \(\omega_1\) in the sense that any such set is Tukey equivalent to one of the five. The author inspects basis problems in a variety of areas, considering successively binary relations on \(\omega_1\), transitive relations on \(\omega_1\), uncountable linear orderings and uncountable regular spaces. He also discusses related Ramsey-theoretic principles such as the Open Coloring Axiom or special Aronszajn trees.
Reviewer: P.Matet (Caen)

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03E65 Other set-theoretic hypotheses and axioms
54E45 Compact (locally compact) metric spaces
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