Applications of Strict Π11 predicates to infinitary logic

Abstract

Consider the predicate of natural numbers defined by: where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.