Open sets in computability theory and reverse mathematics

Computability , Reverse Mathematics and Combinatorics : Open Problems

Fix a finite set L and an infinite list of variables v0, v1, v2, . . . , vn, . . .. For m,n ≤ ω, W (L,m, n) is the set of sequences w of elements of L ∪ {vi | i < m} of length n with the property that vi occurs in w for each i < m and the first occurrence of vi is before the first occurrence of vj whenever i < j < m. When m ∈ ω, W (L,m) is ⋃ n∈ωW (L,m, n). W (L) is ⋃ m∈ωW (L,m). When w ∈ W (L,m...

Reverse mathematics, computability, and partitions of trees

Weexamine the reversemathematics and computability theory of a formofRamsey’s theorem in which the linear n-tuples of a binary tree are colored. Let 2<N denote the full binary tree of height!. We identify nodes of the tree with finite sequences of zeros and ones, and refer to any subset of the nodes as a subtree. For positive integers n, let [2<N]n denote the set of all linearly ordered n-tuple...

Free Sets and Reverse Mathematics

Suppose that f : [N]k → N. A set A ⊆ N is free for f if for all x1, . . . , xk ∈ A with x1 < x2 < · · · < xk , f (x1, . . . , xk ) ∈ A implies f (x1 , . . . , xk ) ∈ {x1, . . . , xk }. The free set theorem asserts that every function f has an infinite free set. This paper addresses the computability theoretic content and logical strength of the free set theorem. In particular, we prove that Ram...

Located Sets and Reverse Mathematics

LetX be a compact metric space. A closed setK ⊆ X is located if the distance function d(x,K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x,K) > r is Σ1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic s...

Computability Theory and Foundations of Mathematics