Degrees of incomputability, realizability and constructive reverse mathematics

Abstract

There is a way of assigning a realizability notion to each degree of incomputability. In our setting, we make use of Weihrauch degrees (degrees of incomputability/discontinuity of partial multi-valued functions) to obtain Lifschitz-like relative realizability predicates. In this note, we present sample examples on how to lift some separation results on Weihrauch degrees to those over intuitionistic Zermelo-Fraenkel set theory IZF.

Rethinking the notion of oracle

“…However, the three algebras mentioned above are the main ones used in applications in this section (although many natural and important examples of relative PCAs are known; see e.g. [37,18] and also Section 4.2). For now, let us proceed with the above three algebras in mind.…”

“…This third approach has been taken by the author [19], for example, and is formulated using topos-theoretic notions such as Lawvere-Tierney topology. This approach also originates from the author's attempt [18] to develop a handy proof method to separate non-constructive principles in constructive reverse mathematics (i.e., reverse mathematics based on intuitionistic logic; see [9,10]).…”

Lawvere-Tierney topologies for computability theorists

“…A similar notion for partial multifunctions on N N has been extensively studied, e.g. in [19,29,16,42,21,15], and is known as generalized Weihrauch reducibility. Indeed, Turing reducibility in the above sense is exactly the restriction of generalized Weihrauch reducibility to functions on N.…”

“…Proof. The proof is by a typical recursion trick; see [21]. Suppose for the sake of contradiction that there exists a winning Arthur-Nimue strategy (τ | η) witnessing LLPO 1/ ≤ LT Error 1/ +2 .…”

“…In general, there are many other toposes that are related to computability theory and (effective) descriptive set theory. As mentioned in Kihara [21], any Σ * -pointclass (see [28]) yields a (relative) partial combinatory algebra, which induces a topos. If the pointclass Π 1 1 is used as a seed, a topos corresponding to "the world of Borel measurable mathematics" will be created, and if the pair (Π 1 1 , Π 1 1 ) is used, a topos corresponding to "the world of effective Borel measurable mathematics" will be created.…”