Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

Hilbert's Metamathematical Problems and Their Solutions

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved

Truth, proof and infinity.

SIGLEAvailable from British Library Document Supply Centre- DSC:DX85238 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results

Computability in constructive type theory

We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice’s theorem, the Myhill isomorphism theorem, and the existence of Post’s simple and hypersimple predicates relying on no other axioms such as Markov’s principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type ℕ → ℕ is L-computable.Wir behandeln eine formalisierte und maschinengeprüfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom für synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Post’s simplen und hypersimplen Prädikaten ohne Annahme von anderen Axiomen wie Markov’s Prinzip oder Auswahlaxiomen. Als zweiten Schritt führen wir Berechnungsmodelle ein. Wir geben einen kompakten Überblick über die Definition von verschiedenen Berechnungsmodellen und erklären maschinengeprüfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprüfte Unentscheidbarkeitsbeweise erlaubt. Wir erklären solche Beweise für die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-Kalkül L als sweet spot für die Programmierung in einem Berechnungsmodell. Wir führen ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ N→N L-berechenbar ist

The Borel hierarchy theorem from Brouwer's intuitionistic perspective

Contains fulltext : 72509.pdf (publisher's version ) (Open Access

“TEACHING REAL NUMBERS IN THE HIGH SCHOOL: AN ONTO-SEMIOTIC APPROACH TO THE INVESTIGATION AND EVALUATION OF THE TEACHERS' DECLARED CHOICES”

The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intuItive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic.The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intutive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic