Groundwork for weak analysis

Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. §1. Introduction. The formalization of mathematics within second-order arithmetic has a long and distinguished history. We may say that it goes back to Richard Dedekind, and that it has been pursued by, among others, Hermann Weyl, David Hilbert, Paul Bernays, Harvey Friedman, and Stephen Simpson and his students (we may also mention the insights of Georg Kreisel, Solomon Feferman