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In [8] we have considered a wide class of " well-behaved " reducibil-ities for sets of reals. In this paper we continue with the study of Borel re-ducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of(More)
The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called ∆ 0 α-reductions, and try to find for various natural(More)
We show that if κ is a weakly compact cardinal then the embed-dability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order on the generalized Cantor space κ 2 there is an L κ + κ-sentence ϕ such that the embeddability relation on its models of size κ, which are all trees, is Borel bireducible(More)
We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ 0 α-measurable functions (for every fixed 1 ď α ă ω 1). Moreover , we present some results concerning those Borel functions(More)