Wadge-like degrees of Borel bqo-valued functions

Abstract

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., {0, 1}-valued functions) to Q-valued functions, for a better quasiorder Q. In this article, we consider more general reducibilities on the Q-valued functions and generalise some results of L. Motto Ros in the first direction and of T. Kihara and A. Montalbán in the second direction: Our main result states that the structure of the ∆ 0 α -degrees of ∆ 0 α+γ -measurable Q-valued functions is isomorphic to the ∆ 0 β -degrees of ∆ 0 β+γ -measurable Q-valued functions, and these are isomorphic to the generalized homomorphism order on the γ-th iterated Qlabeled forests.

A syntactic approach to Borel functions: Some extensions of Louveau's theorem

“…Hereafter, we use the symbol φ α to denote • ω α . As the function symbol φ α = • ω α is specified by the fixed point axiom φ β (φ α (t)) = φ α (t) for any β < α, we say that φ α is the α-th Veblen function symbol, see also [21,17], where the symbol s α is used in [21,32] instead of • ω α or φ α .…”

Degrees of incomputability, realizability and constructive reverse mathematics

“…Σ n,n -functions) are closed under composition, and the induced reducibility notion is studied, e.g. by [18]. The pointclass Π 1 1 is the best-known example of a Spector pointclass, and the induced lightface pca obviously yields hyperarithmetical realizability.…”