We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence (...) and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic. (shrink)

A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left computable iff it is the supremum of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable sequences of rational numbers we introduce a non-collapsing hierarchy {Σn, Πn, Δn : n ∈ ℕ} of real numbers. We characterize the classes Σ2, Π2 and Δ2 in various ways (...) and give several interesting examples. (shrink)

D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or $${\Sigma^b_1-IND^{|x|_k}}$$. It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the (...) other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. (shrink)

It is known that Hume’s Principle, adjoined to a suitable formulation of second-order logic, gives a theory which is almost certainly consistent4 and suffices for arithmetic in the sense that it yields the Dedekind-Peano axioms as theorems. While Hume’s Principle cannot be taken as a definition in any strict sense requiring that it provide for the eliminative paraphrase of its definiendum in every admissible type of occurrence, we hold that it can be viewed as an implicit definition of a (...) sortal concept of cardinal number and, accordingly, as being analytic of that concept. This, coupled with the fact that Hume’s Principle so conceived requires a prior understanding only of (second-order) logical vocabulary, is enough to justify regarding the resulting account of the foundations of arithmetic as a form of logicism. Whether this claim can ultimately be sustained is of course still very much a matter of controversy. But here, rather than add to the debate on that issue, I should like to assume that it can be, and on that basis, address some further questions. (shrink)

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey (...) Theorem is not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is (...) recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

Research on language processing has shown that the disruption of conceptual integration gives rise to specific patterns of event-related brain potentials —N400 and P600 effects. Here, we report similar ERP effects when adults performed cross-domain conceptual integration of analogous semantic and mathematical relations. In a problem-solving task, when participants generated labeled answers to semantically aligned and misaligned arithmetic problems, the second object label in misaligned problems yielded an N400 effect for addition problems. In a verification task, when participants judged (...) arithmetically correct but semantically misaligned problem sentences to be “unacceptable,” the second object label in misaligned sentences elicited a P600 effect. Thus, depending on task constraints, misaligned problems can show either of two ERP signatures of conceptual disruption. These results show that well-educated adults can integrate mathematical and semantic relations on the rapid timescale of within-domain ERP effects by a process akin to analogical mapping. (shrink)

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA (...) 0. (shrink)

Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.

In this paper we present a negative solution of counting problems for some classes slightly different from bounded arithmetic (▵ 0 sets). To get the results we study properties of chains of finite automata.

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set (...) theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book (...) is the first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time. The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. (shrink)

The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It (...) was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (shrink)

Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.

Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about (...) numbers, I explain one way it may be possible to extend our view beyond elementary arithmetic to encompass the theory of real numbers. My approach has affinities to the leading ideas of Frege’s own treatment of the reals, although differing in one fundamental way. I attempt, like the HP approach to elementary arithmetic, to obtain the reals very directly by means of abstraction principles without any essential reliance on a theory of sets. This is the most natural way of extending the neo-Fregean position to the reals. (shrink)

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek (...) to explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in (...) terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well-known non-appropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers. (shrink)

The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate. Experimentation shows the "manyness" of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that (...) are not inevitably arbitrary, and conceptual operations can obviously have a status as natural events with psychology. If the elements and conceptual operations involved in mathematical systems were not inherent in physiological process, various primitive discriminations could not be possible. Also, since some calculi have a natural status in a given empirical circumstance, the axioms of others can not be satisfied. Therefore the psychologist when acting as an empirical scientist seeks a calculus having a structure whose elements are isomorphic with natural units of stimulus and response and whose operations are isomorphic with whatever natural processes are involved. Measurement poses a special problem for the empirical scientist. It concerns but a single class of natural qualities and this only in a limited way. The concept of quantity has a natural counterpart but quantity and measurement are not wholly analogous. Measurement is defined, as H. S. Leonard suggests, as a theoretical activity. Measurement theory has a formal structure but empirical end. Measurement hypothesizes about the position of a particular quality within a definite range of qualities. It therefore is beholden to definite empirical restrictions. Some hypotheses-making systems use terms and relations per se as the context and starting point for dealing with discriminable events. Such procedures are 'transcendent." In empirical science, questions are part of problem-solving activity and their reference is naturally restricted. In providing description and explanation, psychological researchers frequently use calculi in a transcendent way. This results in theories that are only quasi-empirical and "half" true. The roles measurement plays in psychology are discussed. Of particular concern are those cases in which the results of measuring or a theory of measurement are used to define the "real" units, or the "real" relations involved in problematic psychological events, and thence to describe and explain behaviors of interest. Metaphysical or ontological usages of measurement sometimes occur. The implication of these arguments with regard to a view of empirical science is discussed. (shrink)

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said (...) to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

In the 10th century, the Brethren of Purity conceived a henological arithmetic which they believed could explain the mathematical structure of the cosmos (as a macroanthropos), and could lead the student to the discovery of the real substance of his own soul (as a micro-cosmos), a discovery which is the first step towards knowledge of metaphysical and theological truth.

The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the (...) framework of Maimon’s philosophy, arithmetical judgments cannot be analytical, nor is arithmetic an objectively necessary science, but rather only subjectively necessary. We should also cast doubt on his claim that we can create real objects from pure concepts of the understanding. (shrink)

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical (...) concepts. Numbers and arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so (...) that nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

We discuss the philosophical status of the statement that (9n – 1) is divisible by 8 for various sizes of the number n. We argue that even this simple problem reveals deep tensions between truth and verification. Using Gillies's empiricist classification of theories into levels, we propose that statements in arithmetic should be classified into three different levels depending on the sizes of the numbers involved. We conclude by discussing the relationship between the real number system and the (...) physical continuum. (shrink)

In a number of works, we have suggested that whereas the ordered field R of real numbers should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In the present chapter, as part of a more general exposition of the absolute arithmetic continuum, we will outline some of the properties of the system of surreal (...) numbers that we believe lend credence to this mathematico-philosophical thesis. We will also provide an overview of No’s rich structure as a simplicity-hierarchical (or s-hierarchical) ordered field that recursively emerges from the interplay between its structure as an ordered field and its structure as a lexicographically ordered full binary tree. Finally, we will draw attention to how properties of the system of surreal numbers considered as an s-hierarchical ordered algebraic structure can be appealed to in conjunction with classical relations between ordered algebraic and geometric systems to help resolve, for the surreal case, one of the purported difficulties that lies at the heart of attempts to bridge the gap between the domains of number and of geometrical magnitude. In particular, we will explain how it is possible that despite the fact that every surreal number, considered as a member of an s-hierarchical ordered field, differs from every other in characteristic individual properties, the absolute (Euclidean) geometrical continuum, which is modeled by the Cartesian space over the ordered field No of surreal numbers, appears as an amorphous pulp of points that display little individuality. (shrink)

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a (...) property (of rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities are aligned with addition, whereas certain functional relations between entities are aligned with division. Similarly, discreteness vs. continuity of quantities is aligned with different formats for rational numbers. These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook (...) analyses revealed semantic alignments for arithmetic word problems, but not for rational numbers. Nonetheless, Russian college students showed semantic alignments both for arithmetic operations and for rational numbers. Since Russian students exhibit semantic alignments for rational numbers in the absence of exposure to examples in school, such alignments likely reflect intuitive understanding of mathematical representations of real-world situations. (shrink)

Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and (...) followers did : “geometrization of arithmetic”..Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him, the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

We show that true first-order arithmetic of the positive integers is interpretable over the real-algebraic structure of Scott's topological model for intuitionistic analysis. From this the undecidability of the structure follows.

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's (...) Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh+tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oplus \beta _2=\tanh \big +\tanh ^{-1}\big )$$\end{document}, although multiplication β1⊙β2=tanh·tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\odot \beta _2=\tanh \big \cdot \tanh ^{-1}\big )$$\end{document}, and division β1⊘β2=tanh/tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oslash \beta _2=\tanh \big /\tanh ^{-1}\big )$$\end{document} do not seem to appear in the literature. The map fX=tanh-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}=\tanh ^{-1}$$\end{document} defines an isomorphism of the arithmetic in X=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}=$$\end{document} with the standard one in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. The new arithmetic is projective and non-Diophantine in the sense of Burgin, 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov, 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz. Treating the above example as a paradigm, we ask what can be said about the set X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} of ‘real numbers’, and the isomorphism fX:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$\end{document}, if we assume the standard form of Newtonian mechanics and general relativity but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if fX≈0.8sinh/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}\approx 0.8\sinh /$$\end{document}. The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with ΩΛ=0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\Lambda =0.7$$\end{document}, ΩM=0.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _M=0.3$$\end{document}. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element 0X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0_{{\mathbb{X}}}$$\end{document} of addition, is nonzero from the point of view of the standard arithmetic of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. This implies fX-1=0X>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{-1}_{{\mathbb{X}}}=0_{{\mathbb{X}}}>0$$\end{document}. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic. (shrink)

IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements (...) of theorems and proofs in the branch considered into those of some preferred system, where the translation must satisfy certain appropriate conditions (interpretation). The problem is relative to the choice of preferred system, as is Hilbert's consistency problem since he required the consistency to be established by particular methods (finitist ones). A finitist interpretation of classical number theory, which has been published in full detail elsewhere, is here described by means of typical examples. Partial results on analysis (theory of arbitrary functions whose arguments and values are the non-negative integers) are here presented for the first time. One of these results is restricted to functions whose values are bounded; its interest derives from the fact that real numbers may be represented by such functions. It is hoped that diverse general observations and comments, which would bore the specialist, may be of help to the general reader. The specialist may find some points of interest in the last two sections of the main text and in the notes following it. (shrink)

We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques (...) employed will be the reflection principles and Gandy forcing. (shrink)

Improving on a result of Arana, we construct an effective family (φ r | r ∈ ℚ ⋂ [0, 1]) of Σ n -conservative Π n sentences, increasing in strength as r decreases, with the property that ¬φ p is Π n -conservative over PA + φ q whenever p < q. We also construct a family of Σ n sentences with properties as above except that the roles of Σ n and Π n are reversed. The latter result allows (...) to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it to analogues of the E-tree at higher levels of the arithmetical hierarchy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for the (...) applicability of arithmetic to matters of empirical fact?These are philosophical questions, part of a "wider puzzle of explaining the link between experience, language, thought, and the world" (18). But answers to them are heavily constrained by technical results in logic and mathematics, and most of Potter's protagonists are either mathematicians (Frege, Dedekind, Ramsey, Hilbert, Gödel) or mathematically trained philosophers (Kant, Russell, Carnap). (Only Wittgenstein fits neither category.) Potter, whose doctorate was in pure mathematics, is an able guide.It is not hard to think of answers to these questions taken singly; what is hard is answering both at the same time. Formalists can easily explain why arithmetic does not depend for its justification on empirical fact—it is just a game with symbols, not responsible to anything beyond its own rules—but they then have trouble explaining why the world of empirical fact is constrained by this game. Empiricists can explain the applicability of arithmetic in just the same way they explain the applicability of physics, but they then must then explain away the apparent a prioricity and necessity of arithmetic. Kant was the first to propose a plausible answer to both questions: arithmetic is both necessary and empirically applicable because it is grounded in the spatio-temporal form of any possible experience.What unites Frege, Dedekind, Russell, Ramsey, Hilbert, and Carnap is that they reject Kant's answer while continuing to seek an answer to his question: how can arithmetic's necessity be reconciled with its applicability? Potter helpfully classifies their projects by the surrogates they substitute for Kant's appeal to the form of spatio-temporal intuition: Frege and Dedekind appeal to thought, Russell to necessary features of the world, Wittgenstein, Ramsey, and Carnap to language, and Hilbert to the necessary structure of our experience of finitary combinations of objects. (Revealingly, the logicists are not grouped together in Potter's taxonomy.) A chapter (or two) is devoted to each of these thinkers, plus Kant and Gödel.This breadth of coverage does not come at the price of superficiality: Potter's chapters are (by and large) accurate, penetrating, and full of interesting insights. I cannot think of another book that offers such clear explanations of the many mathematical concepts and distinctions one must grasp in order to follow debates in the philosophy of arithmetic. The Russell chapters are enhanced by numerous quotations from archival material. The exposition of Hilbert's programme and the implications for it of Gödel's theorems is the best I have seen. The discussions of Wittgenstein's views on arithmetic and of Carnap's syntactical approach also shine. The chapters on Kant and Frege are somewhat weaker; the discussion of Kant, in particular, would have benefited from some engagement with Michael Friedman's views.But the real contribution of Potter's book is its integration of these nine thinkers into a single coherent story. No one will be surprised that Russell's motivations can only be understood in light of Frege's failures, or that many features of Wittgenstein's cryptic work become clearer when seen in light of the specific problems with Russell's system. But there are also some less obvious payoffs: for example, Hilbert's programme is usefully compared with Russell's regressive method (227-8), and Carnap and Russell are shown to face a common dilemma (269, 286).If there is a central theme to Potter's story, it is "how often we have had to reject an [End Page 455] account not for philosophical reasons but for technical ones" (278). Frege's program founders on Russell's paradox, Russell's... (shrink)

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any suborder of the reals of size (...) ω 1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S κ + ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω 2 ). (shrink)

I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ (...) ω1.Here, I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω1 and continuum many piecewise proper families of cardinality ω2. (shrink)

The paper characterizes the second order arithmetic theorems of a set theory that features a recursively Mahlo universe; thereby complementing prior proof-theoretic investigations on this notion. It is shown that the property of being recursively Mahlo corresponds to a certain kind of β-model reflection in second order arithmetic. Further, this leads to a characterization of the reals recursively computable in the superjump functional.

We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for the (...) applicability of arithmetic to matters of empirical fact?These are philosophical questions, part of a "wider puzzle of explaining the link between experience, language, thought, and the world" (18). But answers to them are heavily constrained by technical results in logic and mathematics, and most of Potter's protagonists are either mathematicians (Frege, Dedekind, Ramsey, Hilbert, Gödel) or mathematically trained philosophers (Kant, Russell, Carnap). (Only Wittgenstein fits neither category.) Potter, whose doctorate was in pure mathematics, is an able guide.It is not hard to think of answers to these questions taken singly; what is hard is answering both at the same time. Formalists can easily explain why arithmetic does not depend for its justification on empirical fact—it is just a game with symbols, not responsible to anything beyond its own rules—but they then have trouble explaining why the world of empirical fact is constrained by this game. Empiricists can explain the applicability of arithmetic in just the same way they explain the applicability of physics, but they then must then explain away the apparent a prioricity and necessity of arithmetic. Kant was the first to propose a plausible answer to both questions: arithmetic is both necessary and empirically applicable because it is grounded in the spatio-temporal form of any possible experience.What unites Frege, Dedekind, Russell, Ramsey, Hilbert, and Carnap is that they reject Kant's answer while continuing to seek an answer to his question: how can arithmetic's necessity be reconciled with its applicability? Potter helpfully classifies their projects by the surrogates they substitute for Kant's appeal to the form of spatio-temporal intuition: Frege and Dedekind appeal to thought, Russell to necessary features of the world, Wittgenstein, Ramsey, and Carnap to language, and Hilbert to the necessary structure of our experience of finitary combinations of objects. (Revealingly, the logicists are not grouped together in Potter's taxonomy.) A chapter (or two) is devoted to each of these thinkers, plus Kant and Gödel.This breadth of coverage does not come at the price of superficiality: Potter's chapters are (by and large) accurate, penetrating, and full of interesting insights. I cannot think of another book that offers such clear explanations of the many mathematical concepts and distinctions one must grasp in order to follow debates in the philosophy of arithmetic. The Russell chapters are enhanced by numerous quotations from archival material. The exposition of Hilbert's programme and the implications for it of Gödel's theorems is the best I have seen. The discussions of Wittgenstein's views on arithmetic and of Carnap's syntactical approach also shine. The chapters on Kant and Frege are somewhat weaker; the discussion of Kant, in particular, would have benefited from some engagement with Michael Friedman's views.But the real contribution of Potter's book is its integration of these nine thinkers into a single coherent story. No one will be surprised that Russell's motivations can only be understood in light of Frege's failures, or that many features of Wittgenstein's cryptic work become clearer when seen in light of the specific problems with Russell's system. But there are also some less obvious payoffs: for example, Hilbert's programme is usefully compared with Russell's regressive method (227-8), and Carnap and Russell are shown to face a common dilemma (269, 286).If there is a central theme to Potter's story, it is "how often we have had to reject an [End Page 455] account not for philosophical reasons but for technical ones" (278). Frege's program founders on Russell's paradox, Russell's... (shrink)

We prove that every computably enumerable random real is provable in Peano Arithmetic to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine” can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with (...) the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the Kraft-Chaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle. (shrink)

We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.