Borel-amenable reducibilities for sets of reals

Abstract

We show that if ℱ is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on (ωω) induced by ℱ turns out to look like the Wadge hierarchy (which is the special case where ℱ is the set of continuous functions).

Game representations of classes of piecewise definable functions

“…We will now prove that, as announced in the previous paragraph, the axiom AD W ξ is strong enough 5) to determine (together with BP and DC(R)) the degree-structure induced by D W ξ , or even by any Borel-amenable set of reductions F ⊇ D W ξ (see [10] for a general introduction to such degree-structures). This shows that to study a Borel-amenable reducibility F we just need to assume an axiom which is "of the same level" of F, rather than the seemingly stronger SLO W .…”

“…Therefore for all the terminology and the results about these concepts we refer the reader to [10] -in fact we suggest to keep a copy of that paper while reading that section in order to compare the various results with the combinatorial arguments proposed here. The unique modification is that here we will sometimes consider reductions from some X ⊆ ω ω to ω ω: given such an X, a set of functions F with domain X, and sets…”

“…Toward our goal, we will simply modify the arguments presented in [10] whenever an axiom stronger than AD W ξ was required. We start by proving a lemma (analogous to [ P r o o f. It clearly suffices to prove that there is no ≤ F -descending chain -the equivalence between this statement and well-foundness can be obtained in the usual way using the existence of a surjection j : ω ω F (see [3,Corollary 2.2]).…”

“…Assuming the Axiom of Determinacy AD, or even just the determinacy of the corresponding games G L and G W , Wadge proved that both ≤ L and ≤ W induce well-behaved stratifications of the subsets of ω ω which have turned out to be very useful in various parts of Set Theory (see e.g. [3,10]). …”

“…In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10].…”