Logical analysis of some theorems of combinatorics and topological dynamics

Separable Banach space theory needs strong set existence axioms

“…for all n ∈ N. The construction can be carried out effectively within WKL 0 because, by Lemma 5.8 of [3], the predicates F ⊂ A n and F o ∩ B n (X * ) ∩ C = ∅ are provably in WKL 0 equivalent to Σ 0 1 formulas. Thus the existence of a sequence of finite sets F n | n ∈ N with the mentioned properties is provable in WKL 0 .…”

“…The ongoing study has been carried out in the context of subsystems of second order arithmetic, under the slogan Reverse Mathematics [12,3,23]. We continue this program here by examining the role of strong set existence axioms in separable Banach space theory.…”

The semigroup βℕ and its applications to number theory

“…(Here, "simplest" refers to ease of presentation. The original combinatorial proof is apparently simplest in terms of the logical structure needed to carry out the proof-see [6].) There are, however, other proofs.…”

The reverse mathematics of Hindman’s Theorem for sums of exactly two elements

“…Hindman's Theorem (HT) [7] states that for every coloring c of N with finitely many colors, there is an infinite set H ⊆ N such that all nonempty sums of distinct elements of H have the same color. Blass, Hirst, and Simpson [1] showed that such an H can always be computed in the (ω + 1)st jump of c, and that there is a computable instance of HT such that every solution computes ∅ ′ . By analyzing these proofs they showed that HT is provable in ACA + 0 (the system consisting of RCA 0 together with the statement that ωth jumps exist) and implies ACA 0 over RCA 0 .…”