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Weak Rudin-Keisler reductions on projective ideals. (English) Zbl 1432.03085

Summary: We consider a slightly modified form of the standard Rudin-Keisler order on ideals and demonstrate the existence of complete (with respect to this order) ideals in various projective classes. Using our methods, we obtain a simple proof of Hjorth’s theorem on the existence of a complete \(\mathbf \Pi ^1_1\) equivalence relation. This proof enables us (under PD) to generalize Hjorth’s result to the classes of \(\boldsymbol {\Pi}^1_{2n+1}\) equivalence relations.

MSC:

03E15 Descriptive set theory
03E60 Determinacy principles
03E05 Other combinatorial set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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