The Pennsylvania State University The Graduate School Department of Mathematics A MODEL OF INTUITIONISM BASED ON TURING DEGREES A Dissertation in Mathematics by Sankha Subhra Basu c 2013 Sankha Subhra Basu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 The dissertation of Sankha S. Basu was reviewed and approved* by the following: Stephen G. Simpson Professor of Mathematics Dissertation Advisor Chair of Committee Jan Reimann Assistant Professor of Mathematics John Roe Professor of Mathematics Sean Hallgren Associate Professor of Computer Science and Engineering Svetlana Katok Professor of Mathematics Director of Graduate Studies *Signatures are on file in the Graduate School. ii Abstract Intuitionism is a constructive approach to mathematics introduced in the early part of the twetieth century by L. E. J. Brouwer and formalized by his student A. Heyting. A. N. Kolmogorov, in 1932, gave a natural but non-rigorous interpretation of intuitionism as a calculus of problems. In this document, we present a rigorous implementation of Kolmogorov's ideas to higher-order intuitionistic logic using sheaves over the poset of Turing degrees with the topology of upward closed sets. This model is aptly named as the Muchnik topos, since the lattice of upward closed subsets of Turing degrees is isomorphic to the lattice of Muchnik degrees which were introduced in 1963 by A. A. Muchnik in an attempt to formalize the notion of a problem in Kolmogorov's calculus of problems. iii Contents Acknowledgements vi 1 Introduction 1 1.1 Overview . 1 1.2 Intuitionistic Logic . 1 1.2.1 History . 1 1.2.2 Brouwer's Intuitionism . 2 1.2.3 The BHK-Interpretation . 4 1.2.4 Other Interpretations for Intuitionistic Propositional and Predicate Logic . 5 1.3 Higher-Order Logics and Sheaf Semantics . 8 1.3.1 Higher-Order Logics . 8 1.3.2 Sheaf Semantics for Higher-Order Intuitionistic Logics . 8 1.4 Recursion Theory . 9 1.4.1 Constructive Recursive Mathematics and the Church-Turing thesis . 9 1.4.2 Unsolvability of the Halting Problem and Turing Degrees . 10 1.4.3 Mass Problems . 10 1.5 Our Contribution . 11 2 Sheaf semantics for higher-order intuitionistic logic 12 2.1 Sheaves over a topological space . 12 2.2 Higher-Order Intuitionistic Logic . 39 2.3 An interpretation of higher-order intuitionistic logic . 41 2.4 Sheaves over poset spaces . 49 2.5 K-sets . 52 3 Number systems and sheaves 57 3.1 The Natural Numbers . 57 3.2 The Baire Space . 62 3.3 The Real Numbers . 65 4 A few principles 67 4.1 The principle of the excluded middle . 67 4.2 The weak law of the excluded middle . 68 4.3 Markov's principle . 69 4.4 The law of trichotomy for real numbers . 74 4.5 The weak law of trichotomy for real numbers . 76 iv 4.6 The axiom of countable choice . 76 5 The Muchnik model 87 5.1 Preliminaries . 87 5.2 The Muchnik topos . 89 5.3 Muchnik Reals . 92 References 108 v Acknowledgements I am grateful to my dissertation adviser Professor Stephen Simpson for his guidance throughout the duration of my graduate study at Penn State. This document would not have been possible without his help, support, and the significant amount of time that he invested in me. I am indebted to Professor Mihir Chakraborty of the University of Calcutta. It was he who in- troduced me to the wonderful world of Logic and Foundations of Mathematics through the courses that he taught at the university and during our meetings and discussions. His continued encour- agement and guidance have been invaluable to me. I thank Professor Wim Ruitenburg of Marquette University. The conversations that I have had with him, during and after my two years as a graduate student at Marquette, helped me sustain my interest in logic. I am thankful to Dr. Michael Warren for explaining the basics of topos theory to me during his visits to Penn State and for sending me his personal notes on sheaf theory. I acknowledge the constant support and inspiration from my parents, my first teachers. They have always urged me to succeed and reach for higher goals. Most of all, they instilled the sense of curiosity, early on in my life, that drives me to learn more. Last, but not the least, I thank my dear wife, who herself embarked on a career in academia to be with me during my years of graduate study. It would have been impossible without her love and support. vi Chapter 1 Introduction 1.1 Overview This thesis connects the concept of Intuitionism or Intuitionistic Logic, as conceived by Luitzen Egbertus Jan Brouwer, and Recursion theory, that ever since Alan Turing's seminal work on com- putability of functions, has been an active field of research. Both of these areas of study originated in the early part of the twentieth century. This connection is motivated by the work of another great mathematician of the twentieth century, Andrey Nikolaevich Kolmogorov, and is laid out via the theory of sheaves over a topological space, an active area of research in geometry. In the following paragraphs, we present a brief history and background of each of these areas along with a description of the contribution of the present document to these areas. 1.2 Intuitionistic Logic 1.2.1 History Intuitionism is a constructive approach to mathematics proposed by Brouwer. The philosophical basis of this approach was present in Brouwer's thesis, titled \On the foundations of mathematics", published in 1907. The mathematical consequences were in his later papers, published between 1912-1928. This was a time, when the world of logic was largely dominated by David Hilbert's proposal for an axiomatic foundation of mathematics (1905) and Georg Ferdinand Ludwig Philipp Cantor's set theory (1874). Intuitionism falls under the broad purview of constructivism. Constructivists were among the leading critics of Hilbert's approach and Cantorian set theory. There were however considerable differences among the various constructivist schools. Some of the other constructivist approaches of the time were Finitism, proposed by Thoralf Skolem in 1923; Predicativism, pro- posed by Herman Weyl in 1918; Bishop's Constructive Mathematics, proposed by Errett Bishop in 1967; Constructive Recursive Mathematics, proposed by Andrei Markov in 1950, to name a few. There were many prominent mathematicians of that time among the constructivists, includ- ing Leopold Kronecker (1823-1891), who is sometimes regarded as \the first constructivist", Ren´e Louis Baire (1874-1932), Emile Borel (1871-1956), Nikolai Nikolaevich Lusin (1883-1950) and Jules Henri Poincar´e(1854-1913). To read more about the different schools of constructivism and for a brief history, see [31][Volume I, Chapter 1], [28]. Constructivism and Intuitionism will be treated as synonymous henceforth. 1 1.2.2 Brouwer's Intuitionism The following is quoted from [31][Volume I, Chapter 1, page 4]. \The basic tenets of Brouwer's intuitionism, are as follows. (a) Mathematics deals with mental constructions, which are immediately grasped by the the mind; mathematics does not consist in the formal manipulation of symbols, and the use of mathematical language is a secondary phenomenon, induced by our limitations (when compared with an ideal mathematician with unlimited memory and perfect recall), and the wish to communicate our mathematical constructions with others. (b) It does not make sense to think of truth and falsity of a mathematical statement independently of our knowledge concerning the statement. A statement is true if we have a proof of it, and false if we can show that the assumption that there is a proof for the statement leads to a contradiction. For an arbitrary statement we can therefore not assert that it is either true or false. (c) Mathematics is a free creation: it is not a matter of mentally reconstructing, or grasping the truth about mathematical objects existing independently of us. It follows from (b) that it is necessary to adopt a different interpretation of statements of the form \there exists an x such that A(x) holds" and \A or B holds". In particular, \ A or not A" does not generally hold on the intuitionistic reading of \or" and \not". In agreement with, but not necessarily following from (c), intuitionism permits consideration of unfinishable processes: the ideal mathematician may construct longer and longer initial segments α(0); : : : ; α(n) of an infinite sequence of natural numbers α where α is not a priori determined by some fixed process of producing values, so the construction of α is never finished.” This brings us to the following question. Which objects can be said to exist as (mental) constructions? \Natural numbers are usually regarded as unproblematic from a constructive point of view; they correspond to very simple mental constructions: start thinking of an abstract unit, think of an- other unit distinct from the first one and consider the combination (\think them together"). The indefinite repetition of this process generates the collection N of natural numbers. Once having accepted the natural numbers, there is also no objection to accepting pairs of natural numbers, pairs of pairs etc. as constructive objects; and this permits us to define Z (the integers) and Q (the rationals)in the usual way as pairs of natural numbers and pairs of integers respectively, each modulo suitable equivalence relations. Infinite sequences of natural numbers, integers or rationals may be constructively given by some process enabling us to determine the n th term (value for n) for each natural number n; in particular, one may think of sequences given by a law (or recipe) for determining all its values (terms).