Effectiveness for infinite variable words and the Dual Ramsey Theorem

Abstract

Abstract. We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k, l) and OVW(k, l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2, 2) nor OVW(2, 2) is provable in WKL 0 . These results give partial answers to questions posed by Friedman and Simpson [3].

Open Questions in Reverse Mathematics

“…Slaman [Sla] showed that DRT k can be proved in Π 1 1 -CA 0 . J. Miller and Solomon [MS04] showed that it is not provable in WKL 0 if k ≥ 3 and that it implies ACA 0 for k ≥ 4.…”

“…However, if we want to talk about general topology, the situation is much less clear. Mummert and Simpson [MS04] have proposed the study of general second-countable topological spaces using MF spaces. Every partial ordering P determines an MF space MF(P ) as follows: The points of MF(P ) are the maximal filters in P , and a basis for the topology is composed of the sets N p = {F ∈ MF(P )|p ∈ F } for p ∈ P .…”

“…We say that a partial ordering P defines a proper MF space MF(P ) if for all p, q ∈ P , we have that p ≤ q if and only if N p ⊆ N q , and that if p|q, there exists r ∈ P with N r = N p ∩ N q . The problem with these types of spaces is that deciding whether a partial ordering P defines a proper MF-space is Π 1 2 -complete (which follows from [MS04]). On the other hand, they could still aid in our understanding of where the complexity in the metrization theorem comes from.…”

A computable analysis of variable words theorems

“…First, note that the statement VW(2, 2) is of the form of Lemma 4.7. Let c : 2 <ω → 2 be the computable instance of VW(2, 2) with no low solution constructed by Miller and Solomon [6] or by Theorem 4.3.…”

“…On the lower bound hand, Miller and Solomon [6] constructed a computable instance c of OVW(2, 2) with no ∆ 0 2 solution, and deduced that RCA 0 + WKL does not prove VW (2,2). Indeed, seeing the instance c of OVW(2, 2) as an instance of VW (2,2), and noticing that the jump of a solution to VW(2, 2) gives a solution to OVW(2, 2), one can deduce that c has no low VW(2, 2)-solution.…”