The Consistency of Arithmetic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On Proof Interpretations and Linear Logic
On Proof Interpretations and Linear Logic On Proof Interpretations and Linear Logic Paulo Oliva Queen Mary, University of London, UK ([email protected]) Heyting Day Utrecht, 27 February 2015 To Anne Troelstra and Dick de Jongh 6 ?? - !A ::: - !A - ! ILLp ILLp ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations IL = Intuitionistic predicate logic 6 ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp - !A ::: - !A IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. -
The Logic of Brouwer and Heyting
THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds. -
Finite Type Arithmetic
Finite Type Arithmetic Computable Existence Analysed by Modified Realisability and Functional Interpretation H Klaus Frovin Jørgensen Master’s Thesis, March 19, 2001 Supervisors: Ulrich Kohlenbach and Stig Andur Pedersen Departments of Mathematics and Philosohy University of Roskilde Abstract in English and Danish Motivated by David Hilbert’s program and philosophy of mathematics we give in the context of natural deduction an introduction to the Dialectica interpretation and compare the interpre- tation with modified realisability. We show how the interpretations represent two structurally different methods for unwinding computable information from proofs which may use certain prima facie non-constructive (ideal) elements of mathematics. Consequently, the two inter- pretations also represent different views on what is to be regarded as constructive relative to arithmetic. The differences show up in the interpretations of extensionality, Markov’s prin- ciple and restricted forms of independence-of-premise. We show that it is computationally a subtle issue to combine these ideal elements and prove that Markov’s principle is computa- tionally incompatible with independence-of-premise for negated purely universal formulas. In the context of extracting computational content from proofs in typed classical arith- metic we also compare in an extensional context (i) the method provided by negative trans- lation + Dialectica interpretation with (ii) the method provided by negative translation + ω A-translation + modified realisability. None of these methods can be applied fully to E-PA , since E-HAω is not closed under Markov’s rule, whereas the method based on the Dialectica interpretation can be used if only weak extensionality is required. Finally, we present a new variant of the Dialectica interpretation in order to obtain (the well-known) existence property, disjunction property and other closure results for typed in- tuitionistic arithmetic and extensions hereof. -
Epistemic Modality, Mind, and Mathematics
Epistemic Modality, Mind, and Mathematics Hasen Khudairi June 20, 2017 c Hasen Khudairi 2017, 2020 All rights reserved. 1 Abstract This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as con- cerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the compu- tational theory of mind; metaphysical modality; deontic modality; the types of mathematical modality; to the epistemic status of undecidable proposi- tions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Each essay is informed by either epistemic logic, modal and cylindric algebra or coalgebra, intensional semantics or hyperin- tensional semantics. The book’s original contributions include theories of: (i) epistemic modal algebras and coalgebras; (ii) cognitivism about epistemic modality; (iii) two-dimensional truthmaker semantics, and interpretations thereof; (iv) the ground-theoretic ontology of consciousness; (v) fixed-points in vagueness; (vi) the modal foundations of mathematical platonism; (vii) a solution to the Julius Caesar problem based on metaphysical definitions availing of notions of ground and essence; (viii) the application of epistemic two-dimensional semantics to the epistemology of mathematics; and (ix) a modal logic for rational intuition. I develop, further, a novel approach to conditions of self-knowledge in the setting of the modal µ-calculus, as well as novel epistemicist solutions to Curry’s and the liar paradoxes. -
Subverting the Classical Picture: (Co)Algebra, Constructivity and Modalities∗
Informatik 8, FAU Erlangen-Nürnberg [email protected] Subverting the Classical Picture: (Co)algebra, Constructivity and Modalities∗ Tadeusz Litak Contents 1. Subverting the Classical Picture 2 2. Beyond Frege and Russell: Modalities Instead of Predicates 6 3. Beyond Peirce, Tarski and Kripke: Coalgebras and CPL 11 3.1. Introduction to the Coalgebraic Framework ............. 11 3.2. Coalgebraic Predicate Logic ...................... 14 4. Beyond Cantor: Effectivity, Computability and Determinacy 16 4.1. Finite vs. Classical Model Theory: The Case of CPL ........ 17 4.2. Effectivity, Choice and Determinacy in Formal Economics ..... 19 4.3. Quasiconstructivity and Modal Logic ................. 21 5. Beyond Scott, Montague and Moss: Algebras and Possibilities 23 5.1. Dual Algebras, Neighbourhood Frames and CA-baes ........ 23 5.2. A Hierachy of (In)completeness Notions for Normal Modal Logics . 24 5.3. Possibility Semantics and Complete Additivity ........... 26 5.4. GQM: One Modal Logic to Rule Them All? ............. 28 6. Beyond Boole and Lewis: Intuitionistic Modal Logic 30 6.1. Curry-Howard Correspondence, Constructive Logic and 2 ..... 31 6.2. Guarded Recursion and Its Neighbours ................ 33 6.3. KM, mHC, LC and the Topos of Trees ................ 34 6.4. Between Boole and Heyting: Negative Translations ......... 35 7. Unifying Lewis and Brouwer 37 7.1. Constructive Strict Implication .................... 37 7.2. Relevance, Resources and (G)BI .................... 40 ∗Introductory overview chapter for my cumulative habilitation thesis as required by FAU Erlangen-Nürnberg regulations (submitted in 2018, finally approved in May 2019). Available online at https://www8.cs.fau. de/ext/litak/gitpipe/tadeusz_habilitation.pdf. 1 8. Beyond Brouwer, Heyting and Kolmogorov: Nondistributivity 43 8.1. -
The Unintended Interpretations of Intuitionistic Logic
THE UNINTENDED INTERPRETATIONS OF INTUITIONISTIC LOGIC Wim Ruitenburg Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI 53233 Abstract. We present an overview of the unintended interpretations of intuitionis- tic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and higher order logic without choice principles or not satisfying the right choice sequence properties. We conclude with remarks on the quest for a correct interpretation of intuitionistic logic. §1. The Origins of Intuitionistic Logic Intuitionism was more than twenty years old before A. Heyting produced the first complete axiomatizations for intuitionistic propositional and predicate logic: according to L. E. J. Brouwer, the founder of intuitionism, logic is secondary to mathematics. Some of Brouwer’s papers even suggest that formalization cannot be useful to intuitionism. One may wonder, then, whether intuitionistic logic should itself be regarded as an unintended interpretation of intuitionistic mathematics. I will not discuss Brouwer’s ideas in detail (on this, see [Brouwer 1975], [Hey- ting 1934, 1956]), but some aspects of his philosophy need to be highlighted here. According to Brouwer mathematics is an activity of the human mind, a product of languageless thought. One cannot be certain that language is a perfect reflection of this mental activity. This makes language an uncertain medium (see [van Stigt 1982] for more details on Brouwer’s ideas about language). In “De onbetrouwbaarheid der logische principes” ([Brouwer 1981], pp. -
Arxiv:1311.3168V22 [Math.LO] 24 May 2021 the Foundations Of
The Foundations of Mathematics in the Physical Reality Doeko H. Homan May 24, 2021 It is well-known that the concept set can be used as the foundations for mathematics, and the Peano axioms for the set of all natural numbers should be ‘considered as the fountainhead of all mathematical knowledge’ (Halmos [1974] page 47). However, natural numbers should be defined, thus ‘what is a natural number’, not ‘what is the set of all natural numbers’. Set theory should be as intuitive as possible. Thus there is no ‘empty set’, and a single shoe is not a singleton set but an individual, a pair of shoes is a set. In this article we present an axiomatic definition of sets with individuals. Natural numbers and ordinals are defined. Limit ordinals are ‘first numbers’, that is a first number of the Peano axioms. For every natural number m we define ‘ωm-numbers with a first number’. Every ordinal is an ordinal with a first ω0-number. Ordinals with a first number satisfy the Peano axioms. First ωω-numbers are defined. 0 is a first ωω-number. Then we prove a first ωω-number notequal 0 belonging to ordinal γ is an impassable barrier for counting down γ to 0 in a finite number of steps. 1 What is a set? At an early age you develop the idea of ‘what is a set’. You belong to a arXiv:1311.3168v22 [math.LO] 24 May 2021 family or a clan, you belong to the inhabitants of a village or a particular region. Experience shows there are objects that constitute a set. -
The Continuum Hypothesis, Part I, Volume 48, Number 6
fea-woodin.qxp 6/6/01 4:39 PM Page 567 The Continuum Hypothesis, Part I W. Hugh Woodin Introduction of these axioms and related issues, see [Kanamori, Arguably the most famous formally unsolvable 1994]. problem of mathematics is Hilbert’s first prob- The independence of a proposition φ from the axioms of set theory is the arithmetic statement: lem: ZFC does not prove φ, and ZFC does not prove φ. ¬ Cantor’s Continuum Hypothesis: Suppose that Of course, if ZFC is inconsistent, then ZFC proves X R is an uncountable set. Then there exists a bi- anything, so independence can be established only ⊆ jection π : X R. by assuming at the very least that ZFC is consis- → tent. Sometimes, as we shall see, even stronger as- This problem belongs to an ever-increasing list sumptions are necessary. of problems known to be unsolvable from the The first result concerning the Continuum (usual) axioms of set theory. Hypothesis, CH, was obtained by Gödel. However, some of these problems have now Theorem (Gödel). Assume ZFC is consistent. Then been solved. But what does this actually mean? so is ZFC + CH. Could the Continuum Hypothesis be similarly solved? These questions are the subject of this ar- The modern era of set theory began with Cohen’s ticle, and the discussion will involve ingredients discovery of the method of forcing and his appli- from many of the current areas of set theoretical cation of this new method to show: investigation. Most notably, both Large Cardinal Theorem (Cohen). Assume ZFC is consistent. Then Axioms and Determinacy Axioms play central roles. -
Irving Ezra Segal (1918–1998)
mem-segal.qxp 5/12/99 12:57 PM Page 659 Irving Ezra Segal (1918–1998) John C. Baez, Edwin F. Beschler, Leonard Gross, Bertram Kostant, Edward Nelson, Michèle Vergne, and Arthur S. Wightman Irving Segal died suddenly on August 30, 1998, After the war while taking an evening walk. He was seventy-nine Segal spent two and was vigorously engaged in research. years at the Insti- Born on September 13, 1918, in the Bronx, he tute for Advanced grew up in Trenton and received his A.B. from Study, where he Princeton in 1937. What must it have been like to held the first of the be a member of the Jewish quota at Princeton in three Guggenheim the 1930s? He told me once that a fellow under- Fellowships that he graduate offered him money to take an exam in his was to win. Other stead and was surprised when Irving turned him honors included down. election to the Na- He received his Ph.D. from Yale in 1940. His the- tional Academy of sis was written under the nominal direction of Sciences in 1973 Einar Hille, who suggested that Segal continue his and the Humboldt Award in 1981. At and Tamarkin’s investigation of the ideal theory the University of of the algebra of Laplace-Stieltjes transforms ab- Chicago from 1948 solutely convergent in a fixed half-plane. But, Segal to 1960, he had fif- wrote, “For conceptual clarification and for other teen doctoral stu- reasons, an investigation of the group algebra of dents, and at MIT, a general [locally compact] abelian group was of where he was pro- Irving Segal interest.” And the thesis was not restricted to fessor from 1960 abelian groups. -
1995 Steele Prizes
steele.qxp 4/27/98 3:29 PM Page 1288 1995 Steele Prizes Three Leroy P. Steele Prizes were presented at The text that follows contains, for each award, the awards banquet during the Summer Math- the committee’s citation, the recipient’s response fest in Burlington, Vermont, in early August. upon receiving the award, and a brief bio- These prizes were established in 1970 in honor graphical sketch of the recipient. of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein Edward Nelson: 1995 Steele Prize for and are endowed under the Seminal Contribution to Research terms of a bequest from The 1995 Leroy P. Steele award for research of Leroy P. Steele. seminal importance goes to Professor Edward The Steele Prizes are Nelson of Princeton University for the following ...for a research awarded in three categories: two papers in mathematical physics character- for a research paper of fun- ized by leaders of the field as extremely innov- paper of damental and lasting impor- ative: tance, for expository writing, 1. “A quartic interaction in two dimensions” in fundamental and for cumulative influence Mathematical Theory of Elementary Particles, and lasting extending over a career, in- MIT Press, 1966, pages 69–73; cluding the education of doc- 2. “Construction of quantum fields from Markoff importance, toral students. The current fields” in Journal of Functional Analysis 12 award is $4,000 in each cate- (1973), 97–112. for expository gory. In these papers he showed for the first time The recipients of the 1995 how to use the powerful tools of probability writing, and for Steele Prizes are Edward Nel- theory to attack the hard analytic questions of son for seminal contribution constructive quantum field theory, controlling cumulative to research, Jean-Pierre Serre renormalizations with Lp estimates in the first influence for mathematical exposition, paper and, in the second, turning Euclidean and John T. -
THE PEANO AXIOMS 1. Introduction We Begin Our Exploration
CHAPTER 1: THE PEANO AXIOMS MATH 378, CSUSM. SPRING 2015. 1. Introduction We begin our exploration of number systems with the most basic number system: the natural numbers N. Informally, natural numbers are just the or- dinary whole numbers 0; 1; 2;::: starting with 0 and continuing indefinitely.1 For a formal description, see the axiom system presented in the next section. Throughout your life you have acquired a substantial amount of knowl- edge about these numbers, but do you know the reasons behind your knowl- edge? Why is addition commutative? Why is multiplication associative? Why does the distributive law hold? Why is it that when you count a finite set you get the same answer regardless of the order in which you count the elements? In this and following chapters we will systematically prove basic facts about the natural numbers in an effort to answer these kinds of ques- tions. Sometimes we will encounter more than one answer, each yielding its own insights. You might see an informal explanation and then a for- mal explanation, or perhaps you will see more than one formal explanation. For instance, there will be a proof for the commutative law of addition in Chapter 1 using induction, and then a more insightful proof in Chapter 2 involving the counting of finite sets. We will use the axiomatic method where we start with a few axioms and build up the theory of the number systems by proving that each new result follows from earlier results. In the first few chapters of these notes there will be a strong temptation to use unproved facts about arithmetic and numbers that are so familiar to us that they are practically part of our mental DNA. -
Edward Nelson (1932-2014) 3
EDWARD NELSON (1932-2014) MIKHAIL G. KATZ AND SEMEN S. KUTATELADZE On September 10, 2014 we lost Edward Nelson, one of the most original and controversial mathematical thinkers of our time. Nelson was born on May 4, 1932 in Decatur, Georgia. His father spoke fluent Russian, having spent some time in St. Petersburg in con- nection with issues related to prisoners of war (Nelson 1995 [11]). Nel- son was educated at the University of Chicago. In 1953 he was awarded a master’s degree, and in 1955 he earned a doctorate under the super- vision of Irving Segal, an expert in mathematical methods of quantum mechanics and functional analysis and one of the founders of the theory of C*-algebras. Not surprisingly, Nelson’s early papers were devoted to quantum mechanics. In 1995, Nelson received the Steele prize awarded by the American Mathematical Society, for his 1963 and 1977 papers on applications of probability theory to quantum fluctuations. After graduating, Nelson spent three years at Princeton’s Institute for Advanced Study, and from 1959 until his retirement in 2013, he worked at Princeton University where he became a Professor in 1964. In his memory the university flag was raised at half-mast for three days. In 1975 Nelson was elected to the American Academy of Arts and Sciences, and in 1997, to the National Academy of Sciences of the USA. Princeton University Press published the following six monographs by Nelson: arXiv:1506.01570v1 [math.HO] 4 Jun 2015 (1) Dynamical theories of Brownian motion. Princeton University Press, Princeton, N.J.