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Analysis in Weak Systems

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Analysis in Weak Systems ANALYSIS IN WEAK SYSTEMS ANTONIO´ M. FERNANDES, FERNANDO FERREIRA, AND GILDA FERREIRA Abstract. The authors survey and comment their own work on weak analy- sis. 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 K¨onig'slemma. A brief discussion of the Baire categoricity theorem follows. It is then consid- ered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integra- tion in the stronger system. The paper finishes with three questions in weak analysis. 1. Introduction This paper is the third and last part of a triptych whose two other parts consist of \Techniques in weak analysis for conservation results" [13] and \Interpretability in Robinson's Q" [20]. This triptych of papers mostly surveys the work of the authors in weak analysis. The present paper is representative of our work in this field and it is the way that the authors have found to jointly render homage to Am´ılcarSernadas on the occasion of his 64th birthday (Am´ılcarwas born on April 20, 1952). The influence and scientific leadership of Am´ılcaron his side of logic has been enormous at Instituto Superior T´ecnicoand, indeed, in Portugal. One should also mention his international collaborations, specially with Brazilian logicians. Am´ılcar’sinfluence was also felt in Portugal on the other side, namely in important matters of academic support and institutional collaboration. The authors of this paper are grateful for the chance to present here the final paper of their triptych. They accepted with pleasure the kind invitation of Francisco Dion´ısioand the other organizers of the Festschrift to participate in the conference (via a presentation of Fernando Ferreira) and to make a contribution to this volume. We must also thank them for giving us the opportunity to render a public homage to Am´ılcar. Unfortunately, less than a year after the conference, Am´ılcarpassed away (on February 7, 2017). This paper is dedicated to his memory. Both this paper and the Techniques paper (also written by the three of us) sur- vey our work on the formalization of analysis in weak sub-exponential systems of second-order arithmetic (the theme of this paper) and on the metamathematical 2010 Mathematics Subject Classification. 03F35, 03B30. Key words and phrases. Weak analysis, bounded arithmetic, weak K¨onig'slemma, polytime computability, integration, Baire category theorem. The three authors acknowledge Centro de Matem´atica, Aplica¸c~oes Fundamentais e In- vestiga¸c~aoOperacional (Universidade de Lisboa). The second author also acknowledge the support of Funda¸c~ao para a Ci^encia e a Tecnologia (FCT) [UID/MAT/04561/2013]. The third author is also grateful to FCT [UID/MAT/04561/2013, UID/CEC/00408/2013 and grant SFRH/BPD/93278/2013] and to Large-Scale Informatics Systems Laboratory (Universidade de Lisboa). 1 2 ANTONIO´ M. FERNANDES, FERNANDO FERREIRA, AND GILDA FERREIRA properties of these systems (the theme of the Techniques paper). This line of re- search was inaugurated by F. Ferreira in his doctoral dissertation [14], written in 1988 in The Pennsylvania State University under the supervision of Stephen Simp- son. Three years before, Samuel Buss had submitted to Princeton University a doc- toral dissertation [5] studying weak systems of arithmetic connected to well-known classes of computational complexity (polytime and polyspace computability, for in- stance). Simpson asked F. Ferreira to look into it and see if a conservation result regarding weak K¨onig'slemma could be proved in the weak setting. Of course, the project was motivated by similar results in reverse mathematics according to which 1 the second-order system WKL0 is Π1-conservative over the base system RCA0 (see [26] for this result and for reverse mathematics in general). It was also connected with the following challenge of Wilfried Sieg: Find a mathematically significant subsystem of analysis whose class of provably recursive functions consists only of the computationally \feasible" ones (see the end of [25]). In his dissertation, F. 1 Ferreira re-worked Buss's arithmetical theory S2 in terms of a system of binary b words (finite 0-1 strings), thereby obtaining the first-order system Σ1-NIA. The recast was done not only because the work with binary strings is more congenial with computational complexity, but also because it made the formulation of weak K¨onig'slemma very transparent. The dissertation proceeded model-theoretically and it was soon realized that, as opposed to the usual setting of reverse mathemat- ics, weak K¨onig'slemma has various interesting formulations in the weak setting. It can be formulated by saying that every infinite binary set-tree (i.e., a tree in the range of the second-order variables of the weak system) has an infinite path, but this most natural form turns out to be insufficient for basic reversals (e.g., Theorem 7). It can also be formulated thus: For each bounded formula of arithmetic which defines an infinite binary tree (not necessarily a set in the weak system) there is an infinite path through the tree. The latter formulation of weak K¨onig'slemma b is denoted by Σ1-WKL. Intermediate formulations do exist and are important for weak analysis (see Section 3). Stronger forms of weak K¨onig'slemma can also be 1 envisaged, as it is the case with the principle of strict Π1-reflection discussed in Section 6. Note that all of these different formulations of weak K¨onig'slemma col- lapse over RCA0. F. Ferreira was able to prove that the theory BTFA together with b 0 b Σ1-WKL is Π2-conservative over Σ1-NIA. The corresponding first-order conserva- b tion result is false because BTFA + Σ1-WKL proves bounded collection, and this b form of collection is independent from Σ1-NIA. The matter was finally clarified in b [16], where it was shown that the first-order consequences of BTFA + Σ1-WKL are b exactly the consequences of bounded collection over Σ1-NIA. It was actually shown b b b that BTFA + Σ1-WKL is first-order conservative over the theory Σ1-NIA + BΣ1, b where BΣ1 is the bounded collection scheme. Unsurprisingly, the proof of this result used Harrington's forcing argument (N.B. It is sometimes thought that Har- rington's conservation result can only be proved using a forcing technique whereas, in fact, it can also be proved by pure proof-theoretic means, as it was shown in [18] via a cut-elimination argument). Most of these matters are discussed in our Techniques paper, but in the above we wanted to convey to the reader how the results first appeared. In F. Ferreira's doctoral dissertation there is little of analysis (nevertheless, e.g., the dissertation considers and studies the Heine/Borel theorem for the Cantor ANALYSIS IN WEAK SYSTEMS 3 space). The formalization of the real number system and of continuous real func- tions was only worked out systematically in [11], together with Ant´onioFernandes. Section 2 below outlines this systematization. The section finishes with the proof of the intermediate value theorem in BTFA (Theorem 4). This result entails that Tarski's theory of real closed ordered fields RCOF is interpretable in BTFA. The in- terest of this remark is that F. Ferreira showed in the last section of [11] that BTFA is interpretable in Raphael Robinson's theory of arithmetic Q. We have, therefore, the following interesting relationship between the very weak arithmetical theory Q and the geometric theory RCOF: The first is not interpretable in the second (because Q is essentially undecidable and RCOF is a decidable theory by Tarski's theorem of quantifier-elimination), whereas the second is interpretable in the first. The theo- ries Q and RCOF could not be more different from the metamathematical point of view, and this issue was the theme of the talk of F. Ferreira in the Festschrift (the title of the talk was \Arithmetic and geometry from the formal point of view: some notes, some lessons"). The fact that BTFA is interpretable in Q is prima facie quite amazing. It rests on the work of people like Edward Nelson, Alex Wilkie, Robert Solovay, Petr H´ajekand others on interpretability in Q. This work was surveyed by F. Ferreira and G. Ferreira in the triptych paper \Interpretability in Robinson's Q," where an old unpublished argument of Solovay appeared for the first time. From this body of work it transpires that the interpretability of BTFA in Q is far from being a strange and isolated phenomena. It is a practical application of the rule of thumb according to which a bounded theory of arithmetic, short of proving the to- tality of exponentiation (actually, for the knowledgeable, short of I∆0 + Ω1), must be interpretable in Q. (Alex Wilkie proved that I∆0 + exp is not interpretable in Q, a result discussed in [20], but as far as we know it is still an open question whether I∆0 + Ω1 is interpretable in Q.) For instance, the Interpretability paper shows that a weak theory of analysis associated with polyspace computability (which contains the counting theory of Section 5, where Riemann integration can be developed) is interpretable in Q. We have already disclosed a bit of the structure of this paper. In the next section, we show how to develop the basic notions of analysis in a weak base second-order theory. Section 3 considers various formulations of weak K¨onig'slemma and their relationships with basic theorems of analysis. In Section 4 we briefly discuss the Baire category theorem in the feasible setting.
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