Peano Arithmetic
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Richard Dedekind English Version
RICHARD DEDEKIND (October 6, 1831 – February 12, 1916) by HEINZ KLAUS STRICK, Germany The biography of JULIUS WILHELM RICHARD DEDEKIND begins and ends in Braunschweig (Brunswick): The fourth child of a professor of law at the Collegium Carolinum, he attended the Martino-Katherineum, a traditional gymnasium (secondary school) in the city. At the age of 16, the boy, who was also a highly gifted musician, transferred to the Collegium Carolinum, an educational institution that would pave the way for him to enter the university after high school. There he prepared for future studies in mathematics. In 1850, he went to the University at Göttingen, where he enthusiastically attended lectures on experimental physics by WILHELM WEBER, and where he met CARL FRIEDRICH GAUSS when he attended a lecture given by the great mathematician on the method of least squares. GAUSS was nearing the end of his life and at the time was involved primarily in activities related to astronomy. After only four semesters, DEDEKIND had completed a doctoral dissertation on the theory of Eulerian integrals. He was GAUSS’s last doctoral student. (drawings © Andreas Strick) He then worked on his habilitation thesis, in parallel with BERNHARD RIEMANN, who had also received his doctoral degree under GAUSS’s direction not long before. In 1854, after obtaining the venia legendi (official permission allowing those completing their habilitation to lecture), he gave lectures on probability theory and geometry. Since the beginning of his stay in Göttingen, DEDEKIND had observed that the mathematics faculty, who at the time were mostly preparing students to become secondary-school teachers, had lost contact with current developments in mathematics; this in contrast to the University of Berlin, at which PETER GUSTAV LEJEUNE DIRICHLET taught. -
Infinite Natural Numbers: an Unwanted Phenomenon, Or a Useful Concept?
Infinite natural numbers: an unwanted phenomenon, or a useful concept? V´ıtˇezslav Svejdarˇ ∗ Appeared in M. Peliˇsand V. Puncochar eds., The Logica Yearbook 2010, pp. 283–294, College Publications, London, 2011. Abstract We consider non-standard models of Peano arithmetic and non- -standard numbers in set theory, showing that not only they ap- pear rather naturally, but also have interesting methodological consequences and even practical applications. We also show that the Czech logical school, namely Petr Vopˇenka, considerably con- tributed to this area. 1 Peano arithmetic and its models Peano arithmetic PA was invented as an axiomatic theory of natu- ral numbers (non-negative integers 0, 1, 2, . ) with addition and multiplication as designated operations. Its language (arithmetical language) consists of the symbols + and · for these two operations, the symbol 0 for the number zero and the symbol S for the successor function (addition of one). The language {+, ·, 0, S} could be replaced with the language {+, ·, 0, 1} having a constant for the number one: with the constant 1 at hand one can define S(x) as x + 1, and with the symbol S one can define 1 as S(0). However, we stick with the language {+, ·, 0, S}, since it is used in traditional sources like (Tarski, Mostowski, & Robinson, 1953). The closed terms S(0), S(S(0)), . represent the numbers 1, 2, . in the arithmetical language. We ∗This work is a part of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic. 2 V´ıtˇezslav Svejdarˇ Infinite natural numbers: . a useful concept? 3 write n for the numeral S(S(. -
The Consistency of Arithmetic
The Consistency of Arithmetic Timothy Y. Chow July 11, 2018 In 2010, Vladimir Voevodsky, a Fields Medalist and a professor at the Insti- tute for Advanced Study, gave a lecture entitled, “What If Current Foundations of Mathematics Are Inconsistent?” Voevodsky invited the audience to consider seriously the possibility that first-order Peano arithmetic (or PA for short) was inconsistent. He briefly discussed two of the standard proofs of the consistency of PA (about which we will say more later), and explained why he did not find either of them convincing. He then said that he was seriously suspicious that an inconsistency in PA might someday be found. About one year later, Voevodsky might have felt vindicated when Edward Nelson, a professor of mathematics at Princeton University, announced that he had a proof not only that PA was inconsistent, but that a small fragment of primitive recursive arithmetic (PRA)—a system that is widely regarded as implementing a very modest and “safe” subset of mathematical reasoning—was inconsistent [11]. However, a fatal error in his proof was soon detected by Daniel Tausk and (independently) Terence Tao. Nelson withdrew his claim, remarking that the consistency of PA remained “an open problem.” For mathematicians without much training in formal logic, these claims by Voevodsky and Nelson may seem bewildering. While the consistency of some axioms of infinite set theory might be debatable, is the consistency of PA really “an open problem,” as Nelson claimed? Are the existing proofs of the con- sistency of PA suspect, as Voevodsky claimed? If so, does this mean that we cannot be sure that even basic mathematical reasoning is consistent? This article is an expanded version of an answer that I posted on the Math- Overflow website in response to the question, “Is PA consistent? do we know arXiv:1807.05641v1 [math.LO] 16 Jul 2018 it?” Since the question of the consistency of PA seems to come up repeat- edly, and continues to generate confusion, a more extended discussion seems worthwhile. -
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. -
Gödel's Second Incompleteness Theorem for Σn-Definable Theories
G¨odel’s second incompleteness theorem for Σn-definable theories Conden Chao∗ Payam Seraji† May 3, 2016 Abstract G¨odel’s second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA is Σn+1-definable and T is Σn-sound, then T dose not prove the sentence Σn-Sound(T ) that expresses the Σn-soundness of T . The optimal- ity of the generalization is shown by presenting a Σn+1-definable (indeed a complete ∆n+1-definable) and Σn−1-sound theory T such that PA ⊆ T and Σn−1-Sound(T ) is provable in T . It is also proved that no recursively enumerable and Σ1-sound theory of arithmetic, even very weak theories which do not contain Robinson’s Arithmetic, can prove its own Σ1-soundness. Keywords: G¨odel’s second incompleteness, Σn-definable, Σn-sound, Σn-Sound, strong provability predicate arXiv:1602.02416v3 [math.LO] 30 Apr 2016 1 Introduction G¨odel’s second incompleteness theorem states that for any recursively enumerable and sufficiently strong (say any extension of Peano’s Arithmetic PA) theory T , T 6⊢ Con(T ), ∗Department of Philosophy, Peking University. My correspondence address is: Room 223 of Building 4th, Changchun New Garden, Peking University, P. C. 100872, Peking, CHINA. Email: [email protected]. †Faculty of Mathematical Sciences, University of Tabriz, 29 Bahman Boulevard, P. O. Box 5166617766, Tabriz, IRAN. Email: p [email protected]. 1 where Con(T ) is the arithmetical sentence expressing the the consistency of T (see e.g. [2, 3, 9, 10]). -
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. -
Infinitesimals Via Cauchy Sequences: Refining the Classical Equivalence
INFINITESIMALS VIA CAUCHY SEQUENCES: REFINING THE CLASSICAL EQUIVALENCE EMANUELE BOTTAZZI AND MIKHAIL G. KATZ Abstract. A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number sys- tem. Such an approach can be seen as formalizing Cauchy’s senti- ment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals. June 2, 2021 1. Historical background Robinson developed his framework for analysis with infinitesimals in his 1966 book [1]. There have been various attempts either (1) to obtain clarity and uniformity in developing analysis with in- finitesimals by working in a version of set theory with a concept of infinitesimal built in syntactically (see e.g., [2], [3]), or (2) to define nonstandard universes in a simplified way by providing mathematical structures that would not have all the features of a Hewitt–Luxemburg-style ultrapower [4], [5], but nevertheless would suffice for basic application and possibly could be helpful in teaching. The second approach has been carried out, for example, by James Henle arXiv:2106.00229v1 [math.LO] 1 Jun 2021 in his non-nonstandard Analysis [6] (based on Schmieden–Laugwitz [7]; see also [8]), as well as by Terry Tao in his “cheap nonstandard analysis” [9]. These frameworks are based upon the identification of real sequences whenever they are eventually equal.1 A precursor of this approach is found in the work of Giuseppe Peano. In his 1910 paper [10] (see also [11]) he introduced the notion of the end (fine; pl. -
1 CS Peirce's Classification of Dyadic Relations: Exploring The
C.S. Peirce’s Classification of Dyadic Relations: Exploring the Relevance for Logic and Mathematics Jeffrey Downard1 In “The Logic of Mathematics; an Attempt to Develop my Categories from Within” [CP 1.417-520] Charles S. Peirce develops a scheme for classifying different kinds of monadic, dyadic and triadic relations. His account of these different classes of relations figures prominently in many parts of his philosophical system, including the phenomenological account of the categories of experience, the semiotic account of the relations between signs, objects and the metaphysical explanations of the nature of such things as chance, brute existence, law-governed regularities and the making and breaking of habits. Our aim in this essay is to reconstruct and examine central features of the classificatory system that he develops in this essay. Given the complexity of the system, we will focus our attention in this essay mainly on different classes of degenerate and genuine dyadic relations, and we will take up the discussion of triadic relations in a companion piece. One of our reasons for wanting to explore Peirce’s philosophical account of relations is to better understand how it might have informed the later development of relations as 1 Associate Professor, Department of Philosophy, Northern Arizona University. 2 In what follows, I will refer to Peirce’s essay “The Logic of Mathematics; An Attempt to Develop my Categories from Within” using the abbreviated title “The Logic of Mathematics.” 3 One reason to retain all of the parts of the figures—including the diamonds that represent the saturated bonds—is that it makes it possible to study more closely the combinatorial possibilities of the system. -
Peano 441 Peano
PEANO PEANO On his life and work, see M . Berthelot, "Necrologie," his position at the military academy but retained his in Bulletin . Societe chindque de France, A5 (1863), 226- professorship at the university until his death in 1932, 227 ; J. R. Partington, A History of Chemistry, IV (London, having transferred in 1931 to the chair of complemen- 1964), 584-585 ; and F. Szabadvary, History of Analytical tary mathematics. He was elected to a number of Chemistry (Oxford, 1966), 251 . scientific societies, among them the Academy of F . SZABADVARY Sciences of Turin, in which he played a very active role . He was also a knight of the Order of the Crown of Italy and of the Order of Saint Maurizio and PEANO, GIUSEPPE (b . Spinetta, near Cuneo, Italy, Saint Lazzaro. Although lie was not active politically, 27 August 1858 ; d. Turin, Italy, 20 April 1932), his views tended toward socialism ; and lie once invited mathematics, logic . a group of striking textile workers to a party at his Giuseppe Peano was the second of the five children home. During World War I he advocated a closer of Bartolomeo Peano and Rosa Cavallo . His brother federation of the allied countries, to better prosecute Michele was seven years older . There were two the war and, after the peace, to form the nucleus of a younger brothers, Francesco and Bartolomeo, and a world federation . Peano was a nonpracticing Roman sister, Rosa . Peano's first home was the farm Tetto Catholic . Galant, near the village of Spinetta, three miles from Peano's father died in 1888 ; his mother, in 1910. -
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. -
Notes on Incompleteness Theorems
The Incompleteness Theorems ● Here are some fundamental philosophical questions with mathematical answers: ○ (1) Is there a (recursive) algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is true? ○ (2) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of Peano or Robinson Arithmetic? ○ (3) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of pure (first-order) logic? ○ (4) Is there a complete (even if not recursive) recursively axiomatizable theory in the language of first-order arithmetic? ○ (5) Is there a recursively axiomatizable sub-theory of Peano Arithmetic that proves the consistency of Peano Arithmetic (even if it leaves other questions undecided)? ○ (6) Is there a formula of arithmetic that defines arithmetic truth in the standard model, N (even if it does not represent it)? ○ (7) Is the (non-recursively enumerable) set of truths in the language of first-order arithmetic categorical? If not, is it ω-categorical (i.e., categorical in models of cardinality ω)? ● Questions (1) -- (7) turn out to be linked. Their philosophical interest depends partly on the following philosophical thesis, of which we will make frequent, but inessential, use. ○ Church-Turing Thesis:A function is (intuitively) computable if/f it is recursive. ■ Church: “[T]he notion of an effectively calculable function of positive integers should be identified with that of recursive function (quoted in Epstein & Carnielli, 223).” ○ Note: Since a function is recursive if/f it is Turing computable, the Church-Turing Thesis also implies that a function is computable if/f it is Turing computable. -
Elementary Higher Topos and Natural Number Objects
Elementary Higher Topos and Natural Number Objects Nima Rasekh 10/16/2018 The goal of the talk is to look at some ongoing work about logical phenom- ena in the world of spaces. Because I am assuming the room has a topology background I will therefore first say something about the logical background and then move towards topology. 1. Set Theory and Elementary Toposes 2. Natural Number Objects and Induction 3. Elementary Higher Topos 4. Natural Number Objects in an Elementary Higher Topos 5. Where do we go from here? Set Theory and Elementary Toposes A lot of mathematics is built on the language of set theory. A common way to define a set theory is via ZFC axiomatization. It is a list of axioms that we commonly associate with sets. Here are two examples: 1. Axiom of Extensionality: S = T if z 2 S , z 2 T . 2. Axiom of Union: If S and T are two sets then there is a set S [T which is characterized as having the elements of S and T . This approach to set theory was developed early 20th century and using sets we can then define groups, rings and other mathematical structures. Later the language of category theory was developed which motivates us to study categories in which the objects behave like sets. Concretely we can translate the set theoretical conditions into the language of category theory. For example we can translate the conditions above as follows: 1 1. Axiom of Extensionality: The category has a final object 1 and it is a generator.