On the role of the collection principle for Σ⁰₂-formulas in second-order reverse mathematics

Abstract

Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of BΣ 0 2 .

Pigeons do not jump high

“…D n k : "For every ∆ 0 n k-partition of ω, there is an infinite subset of one of the parts". Cholak, Jockusch and Slaman [3], Mileti [21] and Chong, Lempp and Yang [4], proved that RT 2 2 is equivalent to COH ∧D 2 2 . The interest of such a decomposition comes from the combinatorial simplicity of COH and D 2 2 .…”

Ramsey’s theorem for singletons and strong computable reducibility

“…Of course, d is not computable from c, merely from the Turing jump of c. Thus, while SRT 2 k is not computably reducible to RT 1 k , it is computably equivalent to a kind of ∆ 0 2 version of RT 1 k called D 2 k , which we will not discuss here. (See [6] and [9, Section 3] for thorough explorations of how these principles are related. )…”

“…More generally, we do not know if there is any set X, and any stable coloring of pairs d computable from X, such that for every infinite homogeneous set H for d, X ⊕ H has p-cohesive degree relative to X. Corollary 1. 6 shows that the answer is no if we ask for H itself, rather than X ⊕ H, to have p-cohesive degree relative to X.…”

The metamathematics of Stable Ramsey’s Theorem for Pairs

“…Hence such a condition does not impose additional assumption to the base theory. See Chong, Lempp, and Yang (2010). model would not satisfy RT 2 2 .…”