Faculteit Wetenschappen Vakgroep Zuivere Wiskunde en Computeralgebra The Reverse Mathematics of Elementary Recursive Nonstandard Analysis: A Robust Contribution to the Foundations of Mathematics Sam Sanders Promotoren: Christian Impens Andreas Weiermann Proefschrift voorgelegd aan de Faculteit Wetenschap- pen tot het behalen van de graad van Doctor in de Wetenschappen richting Wiskunde. 1 I dedicate this dissertation to the founding fathers of Reverse Mathematics and Nonstandard Analysis. 3 Contents Preface 5 Chapter I. ERNA and Reverse Mathematics 7 1. Introduction 7 1.1. Introducing ERNA 7 1.2. Introducing Reverse Mathematics 8 2. ERNA, the system 10 2.1. Language and axioms 10 2.2. ERNA and Transfer 17 2.3. ERNA and the Chuaqui and Suppes system 27 3. Mathematics in ERNA 29 3.1. Mathematics without Transfer 29 3.2. Mathematics with Transfer 37 4. Reverse Mathematics in ERNA 42 4.1. A copy of Reverse Mathematics for WKL0 42 4.2. ERNA and Constructive Reverse Mathematics 50 4.3. Reverse Mathematics beyond WKL0 51 Chapter II. Beyond "-δ: Relative infinitesimals and ERNA 63 1. Introduction 63 2. ERNAA, the system 65 2.1. Language and axioms 65 2.2. Consistency 66 3. ERNAA and Transfer 70 3.1. ERNAA and Stratified Transfer 70 3.2. ERNAA and Classical Transfer 72 3.3. Classical vs. Stratified Transfer 74 4. Mathematics in ERNAA 77 5. ERNAA versus ERNA 84 5.1. More Reverse Mathematics in ERNA 85 5.2. Conservation and expansion for ERNA 89 5.3. Intensionality 93 6. Concluding remarks 96 Chapter III. Relative arithmetic 99 Introduction: internal beauty 99 1. Internal relativity 100 2. The reduction theorem 101 3. Approaching Peano Arithmetic 103 4. Reducing Transfer to the reduction theorem 105 4 . Contents 5. Arithmetical truth 106 6. Philosophical considerations 107 Appendix A. Technical Appendix 109 1. Fundamental functions of ERNA 109 2. Applications of fundamental functions 111 Appendix B. Dutch Summary 117 1. Samenvatting 117 Appendix C. Acknowledgments 119 1. Thanks 119 Appendix. Bibliography 121 Bibliography 121 . Contents 5 Preface Reverse Mathematics (RM) is a program in the Foundations of Mathematics founded by Harvey Friedman in the Seventies ([17, 18]). The aim of RM is to determine the minimal axioms required to prove a certain theorem of `ordinary' mathematics. In many cases one observes that these minimal axioms are also equivalent to this theorem. This phenomenon is called the `Main Theme' of RM and theorem 1.2 is a good example thereof. In practice, most theorems of everyday mathematics are 1 equivalent to one of the four systems WKL0, ACA0, ATR0 and Π1-CA0 or provable in the base theory RCA0. An excellent introduction to RM is Stephen Simpson's monograph [46]. Nonstandard Analysis has always played an important role in RM. ([32,52, 53]). One of the open problems in the literature is the RM of theories of first-order strength I∆0 + exp ([46, p. 406]). In Chapter I, we formulate a solution to this problem in theorem 1.3. This theorem shows that many of the equivalences from theorem 1.2 remain correct when we replace equality by infinitesimal proximity `≈' from Nonstandard Analysis. The base theory now is ERNA, a nonstandard extension of I∆0 + exp. The principle that corresponds to `Weak K¨onig'slemma' is the Universal Transfer Principle (see axiom schema 1.57). In particular, one can say that the RM of ERNA+Π1-TRANS is a `copy up to infinitesimals’ of the RM of WKL0. This implies that RM is `robust' in the sense this term is used in Statistics and Computer Science ([25,35]). Furthermore, we obtain applications of our results in Theoretical Physics in the form of the `Isomorphism Theorem' (see theorem 1.106). This philosophical excursion is the first application of RM outside of Mathematics and implies that `whether reality is continuous or discrete is undecidable because of the way mathematics is used in Physics' (see paragraph 3.2.4, p. 53). We briefly explore a connection with the program `Constructive Reverse Mathematics' ([30, 31]) and in the rest of Chapter I, we consider the RM of ACA0 and related systems. In particular, we prove theorem 1.161, which is a first step towards a `copy up to infinitesimals’ of the RM of ACA0. However, one major aesthetic problem with these results is the introduction of extra quantifiers in many of the theorems listed in theorem 1.3 (see e.g. theorem 1.94). To overcome this hurdle, we explore Relative Nonstandard Analysis in Chapters II and III. This new framework involves many degrees of infinity instead of the classical `binary' picture where only two degrees ‘finite' and ‘infinite’ are available. We extend ERNA to a theory of Relative Nonstandard Analysis called ERNAA and show how this theory and its extensions allow for a completely quantifier- free development of analysis. We also study the metamathematics of ERNAA, motivated by RM. Several ERNA-theorems would not have been discovered without considering ERNAA first. I. Preface 7 CHAPTER I ERNA and Reverse Mathematics That through which all things come into being, is not a thing in itself. Tao Te Ching Lao Tse 1. Introduction 1.1. Introducing ERNA. Hilbert's Program, proposed in 1921, called for an axiomatic formalization of mathematics, together with a proof that this axiomati- zation is consistent. The consistency proof itself was to be carried out using only what Hilbert called finitary methods. In due time, many characterized Hilbert's informal notion of ‘finitary’ as that which can be formalized in Primitive Recursive Arithmetic (PRA), proposed in 1923 by Skolem (see e.g. [51]). By G¨odel'ssecond incompleteness theorem (1931) it became evident that only par- tial realizations of Hilbert's program are possible. The system proposed by Chuaqui and Suppes, recently adapted by R¨osslerand Jeˇr´abek, is such a partial realization, in that it provides an axiomatic foundation for basic analysis, with a PRA consis- tency proof ([11], [42]). Sommer and Suppes's improved system allows definition by recursion, which does away with a lot of explicit axioms, and still has a PRA proof of consistency ([49, p. 21]). This system is called Elementary Recursive Nonstandard Analysis, in short ERNA. Its consistency is proved via Herbrand's Theorem (1930), which is restricted to quantifier-free formulas Q(x1; : : : ; xn), usually containing free variables. Alternatively, one might say it is restricted to universal sentences (8x1) ::: (8xn)Q(x1; : : : ; xn): We will use Herbrand's theorem in the following form (see [11] and [49]); for more details, see [8] and [21]. 1.1. Theorem. A quantifier-free theory T is consistent if and only if every finite set of instantiated axioms of T is consistent. Since Herbrand's theorem requires that ERNA's axioms be written in a quantifier- free form, some axioms definitely look artificial. Fortunately, theorems do not suffer from the quantifier-free restriction. As it turns out, ERNA is not strong enough to develop basic analysis (see theorem 1.3) and hence an extension of ERNA is required. In section 2.2, we extend ERNA with a Transfer principle for universal sentences, called Π1-TRANS. In nonstan- dard mathematics, Transfer expresses Leibniz's principle that the `same' laws hold for standard and nonstandard objects alike. The consistency of the extended theory 8 I. ERNA and Reverse Mathematics − ERNA + Π1-TRANS is provable in PRA via a finite iteration of ERNA's consis- tency proof (see theorem 1.58). The theory ERNA + Π1-TRANS has important applications in `Reverse Mathematics', introduced next. 1.2. Introducing Reverse Mathematics. Reverse Mathematics is a pro- gram in Foundations of Mathematics founded around 1975 by Harvey Friedman ([17] and [18]) and developed intensely by Stephen Simpson, Kazuyuki Tanaka and others; for an overview of the subject, see [46] and [47]. The goal of Reverse Mathematics is to determine what (minimal) axiom system is necessary to prove a particular theorem of ordinary mathematics. By now, it is well known that large portions of mathematics (especially so in analysis) can be carried out in systems far weaker than ZFC, the `usual' background theory for mathematics. Classifying theorems according to their logical strength reveals the following striking phenom- enon: `It turns out that, in many particular cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logi- cally equivalent to the theorem' ([46, Preface]). This recurring phenomenon is called the `Main theme' of Reverse Mathematics (see e.g. [45]) and a good instance is the following theorem from [46, p. 36]. 1.2. Theorem (Reverse Mathematics for WKL0). Within RCA0, one can prove that Weak K¨onig'sLemma (WKL) is equivalent to each of the following mathemat- ical statements: (1) The Heine-Borel lemma: every covering of [0; 1] by a sequence of open intervals has a finite subcovering. (2) Every covering of a compact metric space by a sequence of open sets has a finite subcovering. (3) Every continuous real-valued function on [0; 1], or on any compact metric space, is bounded. (4) Every continuous real-valued function on [0; 1], or on any compact metric space, is uniformly continuous. (5) Every continuous real-valued function on [0; 1] is Riemann integrable. (6) The maximum principle: every continuous real-valued function on [0; 1], or on any compact metric space, is bounded, or (equivalently) has a supre- mum or (equivalently) attains its maximum. (7) The Peano existence theorem: if f(x; y) is continuous in the neighbourhood of (0; 0), then the initial value problem y0 = f(x; y), y(0) = 0 has a continuously differentiable solution in the neighbourhood of (0; 0).