An effective proof that open sets are Ramsey

Abstract

Solovay has shown that if O is an open subset of P (ω) with code S and no infinite set avoids O, then there is an infinite set hyperarithmetic in S that lands in O. We provide a direct proof of this theorem that is easily formalizable in AT R0.

Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

“…(b) Unlike GH, the Ellentuck topology is generated by a continuum of analytic (in fact G δ ) sets; a countable effective coarsening of significance has been studied in [7]. 5.…”

Partial impredicativity in reverse mathematics

“…We use the following uniformly effective version of the clopen Ramsey Theorem (the proof of the clopen Ramsey Theorem in ATR 0 is carried out in [6], with another proof given in [1]; the effective bounds for the complexity are given in [5]). THEOREM 7.9 (ATR 0 ).…”

Hindman's theorem: an ultrafilter argument in second order arithmetic

“…Note that in general, if we can prove a theorem in second order arithmetic plus the existence of an idempotent ultrafilter, we might hope to replace the derivation with the theorem above: if we can arrange all "queries" to the idempotent ultrafilter along a countable well-ordering (where later queries can depend on the results of previous ones), we can then iterate the preceding theorem along the ordering, and the final semigroup will then provide answers to all queries in the proof. (Such methods have been used for proofs using nonprincipal ultrafilters with no additional properties; see, for instance, [1]. )…”