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Infinite dimensional perfect set theorems. (English) Zbl 1275.03140

In this nice paper, the author shows that it is independent of \(\mathsf{ZFC}\) whether the following infinite-dimensional perfect set theorem holds (for \(\alpha\) an arbitrary countable ordinal): For every open set \(A \subseteq [2^\omega]^\omega\), if there is an \(A\)-homogeneous set of cardinality \(\aleph_\alpha\), then there is also a nonempty perfect \(A\)-homogeneous set. Here, a set \(H \subseteq 2^\omega\) is called \(A\)-homogeneous if \([H]^\omega \subseteq A\).
This is proved as follows. For one direction, the existence of a set violating the above statement is derived from the existence of an \(F_\sigma\) set \(A' \subseteq [2^\omega]^2\) for which the analogous \(2\)-dimensional version of the perfect set theorem fails: since the existence of such an \(A'\) is consistent with \(\mathsf{ZFC}\) by [W. Kubiś and S. Shelah, Ann. Pure Appl. Logic 121, No. 2–3, 145–161 (2003; Zbl 1022.03026)], this shows that the infinite-dimensional perfect set theorem consistently fails (Corollary 2.3). For the other direction, the author first proves an infinite-dimensional version of Mycielski’s theorem using a suitable notion of meagerness (Corollary 4.7). From this it follows that in suitable Cohen forcing extensions of the ground model, the infinite-dimensional perfect set theorem is true (Theorem 4.10).
Finally, the above independence result is applied in Section 5 to show that it is independent of \(\mathsf{ZFC}\) whether Tukey reductions of the maximal analytic cofinal type can be witnessed by definable Tukey maps (Theorem 5.7).

MSC:

03E15 Descriptive set theory
03E35 Consistency and independence results

Citations:

Zbl 1022.03026
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References:

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