A theoretical constructivisation of mathematical economics

This thesis deals with some problems in mathematical economics, looked at constructively; that is, with intuitionistic logic. In particular, we look at the connection between approximate Pareto optima and approximate equilibria. We then examine the classically vacuous, but constructively nontrivial, problem of locating the exact point where a line segment crosses the boundary of a convex subset of RN. We also prove the pointwise continuity of an associated boundary crossing mapping. Turning to a rather different aspect of the theory, we discuss Ekeland's Theorem giving approximate minima of certain functions, as well as some fundamental notions in related areas of optimisation. The thesis ends with a discussion of some problems associated with the possible constructivisation of McKenzie's proof of the existence of competitive equilibria

Constructive Analysis of Partial Differential Equations

This thesis presents the results produced in the study of weak solutions of the Dirichlet Problem within Errett Bishop's constructive mathematics. It roughly falls into three major parts: a critical analysis of the classical approaches from a constructive point of view (Chapter 2); constructive results on the existence, stability, and maximality of weak solutions (Chapter 5); and related results on the domains discovered during the course of study on the Dirichlet Problem (Chapters 3 and 6). Chapter 1 introduces constructive mathematics, and Chapter 4 is an auxiliary one in which I give two different constructions of a cut-off function

Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part

Constructive Perspectives on Inductive Logic

Constructive (intuitionist, anti-realist) semantics has thus far been lacking an adequate concept of truth in infinity concerning factual (i.e., empirical, non-mathematical) sentences. One consequence of this problem is the difficulty of incorporating inductive reasoning in constructive semantics. It is not possible to formulate a notion for probable truth in infinity if there is no adequate notion of what truth in infinity is. One needs a notion of a constructive possible world based on sensory experience. Moreover, a constructive probability measure must be defined over these constructively possible empirical worlds. This study defines a particular kind of approach to the concept of truth in infinity for Rudolf Carnap's inductive logic. The new approach is based on truth in the consecutive finite domains of individuals. This concept will be given a constructive interpretation. What can be verifiably said about an empirical statement with respect to this concept of truth, will be explained, for which purpose a constructive notion of epistemic probability will be introduced. The aim of this study is also to improve Carnap's inductive logic. The study addresses the problem of justifying the use of an "inductivist" method in Carnap's lambda-continuum. A correction rule for adjusting the inductive method itself in the course of obtaining evidence will be introduced. Together with the constructive interpretation of probability, the correction rule yields positive prior probabilities for universal generalizations in infinite domains.Työssä tutkitaan havaintoja koskevien väitelauseiden totuutta tilanteissa, joissa havaintojen määrällä ei ainakaan tiedetysti ole ylärajaa. Filosofian ja matematiikan alaan kuuluvassa konstruktiivisessa semantiikassa eli merkitysteoriassa lauseiden merkitys määräytyy niiden todennettavuusehtojen perusteella. Äärettömän havaintomaailman tapauksessa todennettavuusehto on hankalasti muotoiltavissa, koska tällaista maailmaa koskevia yleistyksiä ei yleisessä tapauksessa voi todentaa. Tämä on yhteydessä myös induktion ongelmaan, joka koskee päättelyä menneisyyden havainnoista tulevaisuuteen. Induktiivinen päättely ei säilytä totuutta siinä mielessä, että tosista oletuksista tehtävä johtopäätös ei ole tosi loogisella välttämättömyydellä, vaan korkeintaan todennäköisesti tosi. Työssä esitetään Rudolf Carnapin (1891-1970) induktiivisen logiikan sovelluksena, kuinka äärettömiä havaintomaailmoja koskevien lauseiden totuus voidaan muotoilla konstruktiivisten periaatteiden mukaisesti. Kukin ääretön havaintoprosessi on vapaalakinen jono peräkkäisiä havaintoja, joiden muodostaman kokonaisuuden ominaisuuksia ei voida tietää prosessin äärellisissä vaiheissa. Voidaan kuitenkin tietää, vastaako prosessin annettu äärellinen vaihe jonkin ennalta määritellyn havaintojonon äärellistä vaihetta. Näin voidaan määrittää lauseen konstruktiivinen todennäköisyys äärettömille havaintojonoille: se on niiden ennalta määrättyjen havaintojonojen äärellisten vaiheiden todennäköisyyksien raja-arvo, jotka toteuttavat lauseen kussakin äärellisessä vaiheessaan tietystä vaiheesta alkaen. Tämän todennäköisyyskäsitteen ominaisuuksia tutkitaan suhteessa Carnapin esittämään asymptoottisen todennäköisyyden käsitteeseen. Lisäksi työssä tutkitaan mahdollisuutta määrittää todennäköisyys äärettömyydessä eräänlaisten havaintojonojen joukkojen eli ympäristöjen avulla. Tämän todetaan olevan ristiriidassa sen kanssa, että havaintojonoja koskevat lauseet olisivat konstruktiivisesti tosia äärettömyydessä. Induktiivisessa logiikassa lauseiden todennäköisyys määräytyy ns. induktiivisen menetelmän avulla laskettujen todennäköisyyksien mukaan. Ongelma on, että annettuun tilanteeseen parhaiten soveltuvaa induktiivsta menetelmää ei tiedetä. Etenkään ei tiedetä, onko sellainen induktiivinen menetelmä kaikkein paras, joka ei anna lainkaan painoarvoa havaitulle evidenssille esimerkiksi siten että sata havaittua mustaa korppia lisäisi 101. mustan korpin todennäköisyyttä. Työssä käsitellään myös oikean induktiivisten menetelmän valitsemisen ongelmaa ja päädytään siihen, että toisen kertaluvun todennäköisyydet eivät tarjoa tähän ratkaisua. Sen sijaan induktiivisen menetelmän itsensä päivitys annetun evidenssin nojalla tuottaa tietyin reunaehdoin annettua menetelmää paremman induktiivisen menetelmän. Toisin kuin Carnapin alkuperäisessä järjestelmässä, induktiivisen menetelmän päivitys ja konstruktiivinen semantiikka yhdessä mahdollistavat nollasta poikkeavat todennäköisyydet empiirisille yleistyksille (kuten kaikki korpit ovat mustia )

Introduction to Mathematical Logic, Edition 2021

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem

Metalevel and reflexive extension in mechanical theorem proving

In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available.In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T.The thesis is divided into three parts. The first describes the theory FS₀, which has been suggested by Feferman as a candidate for a framework theory suitable for doing meta-theory. The second describes some experiments with FS₀, proving meta-theorems. The third describes an experiment in extending the theory PRA, declared in FS₀, with a reflection facility.More precisely, in the second section three theories are formalised: propositional logic, sorted predicate logic, and the lambda calculus (with a deBruijn style binding). For the first two the deduction theorem and the prenex normal form theorem are respectively proven. For the third, a relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved in building a full Godel style proof predicate for a theory. It is suggested that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it is possible to exploit an observation of Solovay's that the Lob derivability conditions are sufficient to capture the schematic behaviour of a proof predicate. Combining this with a reflection schema is enough to produce a non-conservative extension of PRA, and this is demonstrated by some experiments