SEARCHING FOR AN ANALOGUE OF ATR₀ IN THE WEIHRAUCH LATTICE

Abstract

There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.

The Open and Clopen Ramsey Theorems in the Weihrauch Lattice

“…Our work explores the formalization of the open and clopen Ramsey theorems as multivalued functions and analyzes their position in the Weihrauch lattice (both the open and the clopen Ramsey theorems are known to be equivalent to ATR 0 over RCA 0 , see [26,Section V.9]). Notice that, as already occurred to other principles equivalent to ATR 0 [13,18], there is not a single multivalued function corresponding to the open Ramsey theorem. Actually, in our case, the situation is even more complex than for the open determinacy or the perfect tree theorem, as the two alternatives (homogeneous solution on the open side or homogeneous solution on the closed side) given by the open Ramsey theorem are not mutually exclusive.…”

“…Three choice problems that turned out to be very relevant to calibrate the strength of multivalued functions that corresponds to theorems around ATR 0 (from the point of view of reverse mathematics) are UC N N , C N N , and TC N N . They have been explored in great detail in [18]. It is known that lim (n) < W UC N N for every n (see [4,Section 6]) and indeed lim † ≡ W UC N N , where lim † corresponds to the iteration of lim over a countable ordinal [22].…”

“…Moreover UC N N and C N N are closed under compositional product [4,Theorem 7.3]. In [18] it is proved that Σ 1 1 -UC N N ≡ W UC N N and Σ 1 1 -C N N ≡ W C N N . The fact that UC N N < W C N N follows from the fact that the element of a Σ 1 1 singleton is hyperarithmetic, but the hyperarithmetic functions are not a basis for the Π 0 1 predicates (see [24, Theorems I.1.6 and III.1.1]).…”

“…Recently Marcone [8] raised the question "What do the Weihrauch hierarchies look like once we go to very high levels of reverse mathematics strength?". There have been several works in this direction: Kihara, Marcone, and Pauly [18] have studied several principles, like the (strong) comparability of well-orders, the perfect tree theorem, and the open determinacy theorem; Goh [13,14] analyzed the weak comparability of well-orders and the König duality theorem; Anglès D'Auriac and Kihara [1] dealt with the Σ 1 1 choice on N and variants thereof. Our work explores the formalization of the open and clopen Ramsey theorems as multivalued functions and analyzes their position in the Weihrauch lattice (both the open and the clopen Ramsey theorems are known to be equivalent to ATR 0 over RCA 0 , see [26,Section V.9]).…”

“…Prominent among these are UC N N , C N N , and TC N N (see Section 2.2 for their precise definitions). It is known that UC N N < W C N N < W TC N N (see [18]).…”