Analysis in weak systems

The authors survey and comment their work on weak analysis. They describe the basic set-up of analysis in a feasible second-order theory and consider the impact of adding to it various forms of weak Konig's lemma. A brief discussion of the Baire categoricity theorem follows. It is then considered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integration in the stronger system. The paper finishes with three questions in weak analysis.info:eu-repo/semantics/publishedVersio

Techniques in weak analysis for conservation results

We review and describe the main techniques for setting up systems of weak analysis, i.e. formal systems of second-order arithmetic related to subexponential classes of computational complexity. These involve techniques of proof theory (e.g., Herbrand’s theorem and the cut-elimination theorem) and model theoretic techniques like forcing. The techniques are illustrated for the particular case of polytime computability. We also include a brief section where we list the known results in weak analysis.info:eu-repo/semantics/publishedVersio

Interpretability in Robinson's Q

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson’s theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson’s position.info:eu-repo/semantics/publishedVersio

Classe de Ciências

info:eu-repo/semantics/publishedVersio

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