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

Psychologism and neopsychologism in philosophy of logic

Thesis focuses on psychologism – a philosophical theory according to which the ontological and epistemological foundations of logic and mathematics are our mental states. Neopsychologism is a new set of psychologistic ideas that appeared already in the XXth century and are influenced by new psychology including cognitive science and artificial intelligence. Its central idea is that the main problem of the early psychologism in logic criticized by Husserl and Frege (Willard 1980) is resolved in the contemporary neopsychologistic research.https://www.ester.ee/record=b517884

Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions

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

Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

Pravda a význam: dialektika teorie a praxe

Tarski's semantic conception of truth is arguably the most influential - certainly, most discussed - modern conception of truth. It has provoked many different interpretations and reactions, some thinkers celebrating it for successfully explicating the notion of truth, whereas others have argued that it is no good as a philosophical account of truth. The aim of the thesis is to offer a systematic and critical investigation of its nature and significance, based on the thorough explanation of its conceptual, technical as well as historical underpinnings. The methodological strategy adopted in the thesis reflects the author's belief that in order to evaluate the import of Tarski's conception we need to understand what logical, mathematical and philosophical aspects it has, what role they play in his project of theoretical semantics, which of them hang in together, and which should be kept separate. Chapter 2 therefore starts with a detailed exposition of the conceptual and historical background of Tarski's semantic conception of truth and his method of truth definition for formalized languages, situating it within his project of theoretical semantics, and Chapter 3 explains the formal machinery of Tarski's truth definitions for increasingly more complex languages. Chapters 4-7 form the core of the...Tarského sémantická koncepce pravdy je patrně nevlivnější - určitě nejdiskutovanější - moderní koncepce pravdy, která vzbudila nespočet různých interpretací a reakcí. Zatímco někteří filosofové ji oslavovali jako úspěšnou explikaci pojmu pravdy, jiní argumentovali, že nám neposkytuje adekvátní filosofický výklad tohoto pojmu. Cílem dizertace je podat systematické a kritické prozkoumání povahy a signifikance Tarského koncepce, založené na pečlivé expozici jejich konceptuálních, technických I historických předpokladů. Metodologická strategie aplikována v práci obráží autorovo přesvědčení, že nelze patřičně zhodnotit přínos Tarského koncepce bez pochopení jejich logických, matematických a filosofických aspektů, a toho jakou roli hraji v jeho širším projektu teoretické sémantiky, jak spolu souvisí (případně nesouvisí). Kapitola 2 je detailní expozicí konceptuálního i historického pozadí Tarského koncepce pravdy a metody definovaní pojmu pravdy pro formalizované jazyky, a v kapitole 3 se vysvětluje formální aparát pravdivostních definicí pro 3 typy jazyků různé komplexity. Kapitoly 4-7, které tvoří jádro celé práce, jsou věnovány ústřední otázce signifikance Tarského koncepce. V kapitole 4 se vysvětlují její logicko-matematické aspekty a přínos pro matematickou logiku, v souvislosti s výsledky Kurta...Institute of Philosophy and Religious StudiesÚstav filosofie a religionistikyFilozofická fakultaFaculty of Art

The Normativity of Logic in a Psychologistic Framework: Three Approaches

Contemporary psychologism has been amended for most of the objections by its opponents over a century ago. However, some authors still raise doubts about its ability to account for some peculiar properties of logic. In particular, it is argued that the psychological universality of patterns of inferential behavior is not sufficient to account for the normativity of logic. In this paper, I deal with the issue and offer three alternative solutions that do not rely on mere empirical universality. I will use the works of Laurence Jonathan Cohen, Diego Marconi and Marcello D'Agostino, adapting them for the purpose of defending logical psychologism. I will therefore argue that, although more refined work on the subject is needed, contemporary psychologism has the key resources to retain its place in the philosophical debate on the foundations of logic

Теорија модела и егзактност научног представљања

Модели играју централну улогу у многим начним контекстима, што чини теорију модела једном од фокалних тачака савремене филозофије науке. Тема ове докторске дисертације је постепено конституисање теорије модела током последњих пола века, почевши са напуштањем програма логичког позитивизма и пионирским радовима Супеса и других аутора. Семантички приступ научним конструктима, који је наследио доминантну позицију у савременом разумевању науке, резултовао је у оштром истицању модела присутних у науци. Ипак, реконструкција научних конструката и научног резоновања помоћу математике, која је уобичајена алатка семантичког приступа, производи многе потешкоће, посебно оне везане за нучну праксу и примену научних теорија. Ја ћу тврдити да, упркос значајним предностима добивеним семантичком моделском ревоулицијом, реконструисање науке формалним средствима је у крајњој линији узалудно, првенствено зато што анализа научне праксе открива да се, чак и у најегзактнијим наукама, математика употребљава на ограничен и апроксимативан начин, погодно прилагођен специфичним научним потребама датог случаја. Ослањајући се на анализе које откривају импровизирајући карактер модела и науке уопште, покушаћемо заправо да ојачамо схватање модела као главних сазнајних и представљачких инструмената модерне науке. Главни циљ ове дисертације биће да обједини разнолике критике семантичке ортодоксије, у циљу да пружи јединствени приступ научног моделовања усмерен на праксу. Ово ћемо покушати да постигнемо путем развијања приступа заснованог на појму сличности, истовремено чувајући с моделима повезане увиде семантичког приступа и примењујући функционалистичку анализу модела. Овим методом, надамо се, дистанцираћемо се од ограничавајућих формализама и развити оквир способан да прихвати и научну когнитивну реалност, наиме њен ''људски аспект'', и њену методолошку отворност коју толико ценимо...Models play a central role in many scientific contexts, which makes model theory one of the focal points of the contemporary philosophy of science. The topic of this doctoral thesis is the gradual constitution of model theory during the last half-century, starting with the abandonment of the logical positivism program and the pioneering works of Suppes and others. The succeeding semantic approach, which resulted in emphasis of scientific models, grew to the dominant contemporary position in understanding science. However, reconstruction of scientific constructs and scientific reasoning by mathematics, the common tool of the semantic approach, spawns many difficulties, specifically the ones related to the scientific practice and the application of scientific theories. I will argue that, in spite of the significant advantages gained with the semantic model-revolution, reconstructing science by formal tools is futile, most notably on the grounds that practice-analysis reveals that, even in the most exact sciences, mathematics is used in limited and approximating ways, suitably adapted to the specific scientific needs at hand. We‘ll build on the analyses revealing the improvising character of models and science in general, and try to strengthen the conception of models as the main cognitive and representational instruments of modern science. Our main goal will be to unify diverse critiques of the semantic orthodoxy, in order to give a unitary, practice-oriented account of modeling in science. We'll try to accomplish this by evolving similarity-based approach in such a way as to preserve model related insights of the sematic approach and by employing a functionalistic analysis of models. This way, we hope, we'll distance ourselves from the restrictive formalisms and develop a framework able to accommodate both scientific cognitive reality, notably its human aspect, and her methodological openness we so much cherish...