The Riemann Integral in Weak Systems of Analysis

Fernando Ferreira (Universidade de Lisboa, Portugal)

Gilda Ferreira (CMAF-Universidade de Lisboa, Portugal)

Abstract: Taking as a starting point (a modification of) a weak theory of arithmetic of Jan Johannsen and Chris Pollett (connected with the hierarchy of counting functions), we introduce successively stronger theories of bounded arithmetic in order to set up a system for analysis (TCA²). The extended theories preserve the connection with the counting hierarchy in the sense that the algorithms which the systems prove to halt are exactly the ones in the hierarchy. We show that TCA² has the exact strength to develop Riemannian integration for functions with a modulus of uniform continuity.

Keywords: Riemann integral, counting hierarchy, weak analysis

Categories: F.1.3, F.4.1