Basic Applications of Weak König’s Lemma in Feasible Analysis

In the context of a feasible theory for analysis, we investigate three fundamental theorems of analysis: the Heine/Borel covering theorem for the closed unit interval, and the uniform continuity and the maximum principles for real valued continuous functions defined on the closed unit interval. §1. The three results. The business of reverse mathematics is to investigate the logico-mathematical strength of the various theorems of ordinary mathematics. This investigation is usually carried over the second-order base theory RCA0 – a theory whose proof-theoretic strength is that of primitive recursive arithmetic. In this article, we investigate three basic theorems of analysis over a feasible base theory, i.e., a theory whose provably total functions (with appropriate graphs) are the polynomial time computable functions. Our feasible base theory is BTFA, a theory introduced by Ferreira in a paper entitled “A feasible theory for analysis” [8]: we presuppose familiarity with the notation and results of that paper and an acquaintance with the basic features of research in reverse mathematics (as exposed in the relevant sections of chapters II, III and IV of [10]). Notice that the first-order part of the intended model of BTFA is 2, the set of finite sequences of zeros and ones (also called binary words or strings), as opposed to the more traditional setting of the natural numbers. As it happens, we find the binary setting more perspicuous for dealing with theories concerned with sub-exponential classes of computational complexity. The first-order part of a model of BTFA is denoted by W (for words). Given a formula A of the language of BTFA and x a distinguished (firstorder) variable, we say that A defines an infinite subtree of W, and write Tree∞(Ax), if ∀x∀y(A(x) ∧ y ⊆ x → A(y)) ∧ ∀n ∈ T∃x( (x) = n ∧A(x)), 1 2000 Mathematics Subject Classification. 03B30, 03F35.