Dedekind Cut

Dedekind Cut Student: Arturo Domingo Valero Botella, Unidersidad de Barcelona Teacher: Paolo Maffezioli, Development of Formal Logic Abstract In this research I have tried to summarize the cuts of Dedekind in the most concise way possible. I begin with a historical instruction with which I intend to place Dedekind’s works in the history of mathematics. To introduce the concept of cutting, which is the central concept of my research, I begin by stating how the cuts are incomplete for rational numbers and once these concepts are settled I try to move towards the completeness of the cuts in real numbers and how they are intimately linked to the concept of continuity. I also explain as an introduction how we can operate with the cuts. Resumen En este trabajo de investigación he intentado resumir las cortaduras de Dedekind de la manera mas concisa posible. Comienzo con una intruduccion historica con la que pretendo situar los trabajos de Dedekind en la historia de las matematicas. Para introducir el concepto de cortadura, que es el concepto central de mi investigación, comienzo exponiendo como las cortaduras son incompletas para los números racionales y una vez asentado estos conceptos intento avanzar hacia la completitud de las cortaduras en los números reales y como estan intimamente ligadas al concepto de continuidad. Además expongo a modo de introducción como podemos operar con las cortaduras. Keywords— Dedekind, Dedekind’s cut, cut, real numbers, rational numbers, continuity, Mathematical logic Palabras clave— Dedekind, Cortaduras de Dedekind, Números reales, Números racionales, Cotinunidad, Lógica matemática 1 1. Introduction A criticism from its beginnings in mathematics a true lack of foundations for this discipline, a truly scientific foundation. In his opinion, the mathematicians of his time continue in the Euclidean paradigm of geometric intuition, and we have not given way to a place pure algebraic mathematics, which was the path discovered by Descartes and defended by Leibniz among others. Dedekind admits that geometric intuition is fundamental in the first teaching of differential calculus, since teachers begin with algebra it would take more time for students to know the basics . But what was not accepted was to use geometry (implicitly or explicitly) in the most rigorous exposures of differential calculus. He was the first to clearly and concisely expose a scientific and arithmetic basis of both natural and real numbers. Consider arithmetic in its entirety as the same operation of the same act of counting, and we will count not on the infinite succession of natural numbers, rational numbers and negatives are nothing more than a consequence of mathematical signs with natural numbers. Before him, and bad math teachers still do, irrational numbers were defined as those that are rational to us, and real numbers as the union of rational and rational numbers, and this definition is far from being a true definition Scientific and axiomatic. In the text ’Continuity and irrational numbers’ he states that the body of rational numbers is a closed and complete body, but lacks continuity. In other words, if we compare the line of the rational numbers with a continuous line, we perceive that there are an infinity of points on the line that do not have their correspondence in the rational numbers. In other words, there is an infinity of immeasurable lengths with the unit of length, as the ancient Greeks already knew. But the greatness of Dedekind is in its concept of cut, which mathematicians 2 now call body. The bases are based on the starting set, that is, the set is closed under operations. And this property only fully meets the real numbers and the property of continuity. For this reason the rational numbers are incomplete and discontinuous, but as we will show later, the class of all cuts in the discontinuous domain of the numbers. I would like to emphasize that Dedekind was the first mathematician to arrive at an approximation of pure set theory. In ”Labyrinth of thought”, Jose Ferreiros referred to this singular origin of set theory with arguments and data that attest to the importance of the origin of the theory and in the first phase of its development as Richard Dedekind (1831-1916) . As Ferreiros says, he is not the first to call attention to the role played by Dedekind in this regard: Zermelo himself, when in 1908 he pressed the first axiomatization of set theory, he referred to it as a theory created by Cantor and Dedekind . 2. Rational numbers A number of the form p q is a rational number,where p,q are integers; q 6= 0; q > 0 and p and q have no common factor. We know that when we have two rational numbers, then addition, subtraction, multiplication, division (denominator is not equal to zero), the result is also a rational number. So if we apply the addition, that is if two rational numbers and addition, we get again a rational number, and the same for all the operations. So basically, it satisfy all the algebraic. The set of rational numbers is an Ordered field. 2.1. Ordered fields We mean that if we pick up any two rational number from the collection of rational numbers, then: 3 Comparable: one can easily identify which number is greater than the other or which number is lower, less than the other. or they are, or are equal Transitivity: α1 < α2 α2 < α3 , then α1 < α3 . Between any two rational numbers say α and β there exists a rational number lying between and β: α< α+β <β 2 i.e., Between any two rarional numbers, one can introduce infinite number of rational points. But those points which are not at all rational points, we cannon be put it in the set of rational number. So it means, the set of rational numbers is not all Complete. So we need the further extension of the rational point. Now for example, if we take a point under square root of two. We claim that square root of two is not a rational number (16:40) 2.2. Cuts of rational numbers We can divide the positive rational numbers into two classes, one containing the numbers whose squares are less than 2, and the other those whose squares are greater than 2. These classes have some properties: 1. Every member of R (upper class) is greater than every member of L. 2. These is a member of L, and a member of R, whose difference is as small as we please. 3. The class L has no greatest number and the class R has no least number √ (in case q = 2). i.e., the class L which contains all rationals numbers 4 whose square is ¡ 2, and the class R ¿2. For these classes property (3) holds Proof: 41:20 It can be constructed with reference to any two properties, say P and Q which are mutually exclusive and one of which is essentially possessed by every rational number. e.g., L: set q rational number and the set of rational numbers x which are less than 1, R: set q rational number and the set of rational numbers x which are not less than 1, An we can define the numbers as the cut of these classes 1= (L,R) 3. Real numbers The mode of division of numbers into two classes is known as section or cut of numbers. In general,the section of rational numbers can be constructed with the help of any two properties say P and Q which are mutually exclusive and one of which is essentially possessed by every rational number. If every numbers which possesses the property P, less than every number which possesses the property Q, then former?? Will constitute the class denoted by L and later will constitute the class denoted by R. L = {x ∈ Q : x ≤ 1} R = {x ∈ Q : x > 1} Since cut rational numbers are of there types 1. The class L has a greatest number say l and the class R has no least number. 5 2. The class R has a least number say r and the class L has no greatest number. 3. The class L has no greatest number and the class R has no least number. Type 1. and 2. corresponds to rational numbers l and r respectively. While type 3. give a irrational numbers. Thus any real number α can be defined of section (L,R). We know by Dedekind that since the aggregate of sections of rational numbers is larger aggregate than that q rational numbers themselves. Thus numbers corresponding to these additional sections are defined as irrational numbers 3.1. Dedekind Theorem If the system of real numbers is divided into two belongs to classes L and R in such a way that 1. Each class contains at least one number. 2. Every number belongs to one class or the other. 3. Every number in the lower class L is less than every number in the upper class R. Then there is a number α such that every number less than α belongs to L and every number greater than α belongs to R. The number α itself may belong to either class. 3.2. Definition of operators using cuts Let α1 and let α2 be two real numbers Given by the cut (L1 , R1 ) and (L2 , R2 ) respectively, there are 3 possibilities 6 Every member of L1 belongs to L2 and every member of R1 belongs to R2 . That is the equality (α1 = α2 ). Every member of L1 belongs to L2 but every member of R1 does not belong to R2 . That is less than (α1 < α2 ). Every member of L1 does not belongs to L2 , but every member of R1 belongs to R2 . That is great than (α1 > α2 ). The real number α given by the cut (L,R): Positive real number when the lower class L contains some positive numbers, i.e., L will contains all negative, 0 and some positive. Negative real number α when the upper class R contains some negative numbers If all the numbers in L are negative and all the numbers in R are positive, the real number defined by the cut is 0. −α = (−L1 , −R1 ). 1 α = ( L11 , R11 ). − α1 = 1 −α The addition α1 + α2 is cut correspond to α3 = (L3 , R3 ), such that, any number a3 ∈ L3 can be written as a3 = a1 +a2 , where a1 ∈ L1, a2 ∈ L2 for some number. Similary, b3 ∈ R3 can be written as b3 = b1 + b2 , where theb1 ∈ R1 , b2 ∈ R2 . And the multiplication: If (L1 , R1 ) and (L2 , R2 ) are both non-negative, then we define (L1 , R1 ) × (L2 , R2 ) to be the pair (L3 , R3 ) where L3 is the set of all products a1 a2 where a1 L1 and a2L2 and at least one of the two numbers is non-negative. R3 is the set of all products of the form b1 b2 where b1 ∈ R1 and b2 ∈ B2 . 7 If (L1 , R1 ) is the 0 cut, then (L1 , R1 ) × (L2 , R2 ) = (L1 , R1 ). If (L2 , R2 ) is the cut 0, then (L1 , R1 ) × (L2 , R2 ) = (L2 , R2 ). If (L1 , R1 ) is negative and (L2 , R2 ) is positive, then (L1 , R1 ) × (L2 , R2 ) = −(−(L1 , R1 ) × (L2 , R2 )). If (L1 , R1 ) is positive and (L2 , R2 ) is negative, then (L1 , R1 ) × (L2 , R2 ) = −((L1 , R1 ) × −(L2 , R2 )). If (L1 , R1 ) is negative and (L2 , R2 ) is negative, then (L1 , R1 ) × (L2 , R2 ) = (−(L1 , R1 ) × −(L2 , R2 )). From the addition and multiplication we can define the rest of arithmetic operations 4. Natural numbers As a curiosity and in order to show all the work that Dedekind contributed to arithmetic, it is known that Peano’s axioms of natural numbers were first defined by Dedekind in ’What are numbers and what should they be?’ (1888). Peano just populated them. 5. References R. Dedekind, ’Continuity and irrational numbers’ and the letters of Dedekind to R. Lipschitz (1872), translation by Juan Climent and Juan Dedios Bares (Uviversidad de Valencia). https://www.uv.es/ jkliment/ Lorena Zafra Granados, ¿Euclides es a proporción como Dedekind es a cortaduras? (2012). ://funes.uniandes.edu.co/11851/1/Zafra2012Euclides.pdf 8 Stanford Encyclopedia of Philosophy,’Dedekind’s Contributions to the Foundations of Mathematics’ (Oct 28, 2016). https://plato.stanford.edu/entries/dedekind-foundations/FouAri Ignacio Jané, ’Jose Ferreiros, Labyrinth of Thought. A history of Set Theory and its Role in Modern Mathematics.’ (1999). https://web.archive.org/web/20101011011205/http://divulgamat.ehu.es/weborriak/ historia/Gaceta/Historia43.pdf 9