WHY A DEDEKIND CUT DOES NOT PRODUCE IRRATIONAL NUMBERS And, Open Intervals are not Sets Pravin K. Johri The theory of mathematics claims that the set of real numbers is uncountable while the set of rational numbers is countable. Almost all real numbers are supposed to be irrational but there are few examples of irrational numbers relative to the rational numbers. The reality does not match the theory. Real numbers satisfy the field axioms but the simple arithmetic in these axioms can only result in rational numbers. The Dedekind cut is one of the ways mathematics rationalizes the existence of irrational numbers. Excerpts from the Wikipedia page “Dedekind cut” A Dedekind cut is а method of construction of the real numbers. It is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. The countable partitions of the rational numbers cannot result in uncountable irrational numbers. Moreover, a known irrational number, or any real number for that matter, defines a Dedekind cut but it is not possible to go in the other direction and create a Dedekind cut which then produces an unknown irrational number. 1 Irrational Numbers There is an endless sequence of finite natural numbers 1, 2, 3 … based on the Peano axiom that if n is a natural number then so is n+1. The real numbers are points on the real line and satisfy the Field axioms. Field Axioms 0, 1 are real numbers and the following exist: A unique additive inverse -x for any real number x 0 -1 A unique multiplicative inverse x , for any real number x 0 Real numbers x + y and x * y, for every two real numbers x, y. Along with associative, commutative, and distributive properties. Real numbers can be rational or irrational. The sets N, Q and R are the infinite sets of natural, rational and real numbers, respectively. The theory says that N and Q are countable while R is uncountable, a higher order of infinity, and almost all real numbers are irrational. Further, real numbers can be represented in the infinite decimal notation and irrational numbers are written to infinite non-repetitive decimal digits. This is self-contradictory because The decimal formula applies only to fixed finite decimal digits. Otherwise, the arithmetic in the formula is no longer valid. The Cantor Diagonal Argument (CDA) does not examine numbers to infinite digits even though there is an explicit requirement in the CDA that all numbers are written to infinite digits (see Appendix D). This requirement in the CDA is unnecessary and does nothing! The number 1/3 = 0.333… with repeating digits is considered to be written to infinite digits. Repetition results in an unending potential infinite (see Appendix C). Even if it is assumed it can be carried out to “completion”, whatever that means as it is not defined precisely, and one actually ends up with infinite decimal digits, it so turns out that such numbers are also rational. 2 All numbers written to finite digits are rational. The irrational numbers do not fit the repetitive “infinite” representation. The only option left is that irrational numbers have actual infinite digits but the actual infinite has a nonfinite or transfinite magnitude. Arithmetic cannot be carried out with non-numbers. Even under the assumption there can be infinite decimal digits, no number can be written to non-repetitive infinite digits and, out of necessity, irrational numbers are denoted with symbols like e, π and 2 leading to the biggest absurdity in mathematics: Almost all real numbers cannot be written as numbers! There are very few examples of the irrational numbers and yet they combine to produce a higher-order, uncountable infinity. According to the theory, the CDA establishes that the unit interval is uncountable (although this is not right - see Appendix D). There is a second infinity, boundlessly larger than the first. The uncountability of R would be a profound observation were it not contradicted by another result - the rational numbers are dense in the real numbers! Theorem 1: (Density of the Rationals): Given any two real numbers a, b with a < b, there exists a rational number q such that a < q < b. Archimedean property: Given any real number x > 0 there is a natural number n such that n > x-1 Proof: The proof is based on the Archimedean property. Since b – a > 0, there is a n such that n > (b – a)-1 and n a + 1 < n b Let k be the largest natural number such that k n a and k + 1 > n a Combining the inequalities n a < k + 1 n a + 1 < n b And q = (k + 1) / n is a rational number in (a, b) This is a constructive proof in that q is derived from a and b. 3 The countable rational numbers are densely distributed in the uncountable real numbers. There is a rational number between every two real numbers, which themselves could be rational or irrational leading to the commonsense inference that At least half of the real numbers must be rational. There is a theorem that the irrational numbers are also dense in the real numbers but this is to be expected as the irrational numbers are supposed to be vastly more numerous. Theorem 2: (Density of the Irrationals): Given any two real numbers a, b with a < b, there exists an irrational number p such that a < p < b. Proof: When a and b are both positive: From theorem 1, there exists a rational number q in the interval (a/2, b/2). 2q must be irrational because, if 2q is rational, then 2 is also rational. Hence, p = 2q (a, b) This proof relies on a pre-existing irrational number such as 2. Mathematics does not explain how the irrational and rational numbers lie on the real line in accordance with these two theorems. What is abundantly clear is that on the real line, There cannot be an interval with just rational numbers in it, and There cannot be an interval with just irrational numbers in it. Which, taken together, yields the commonsense inferences The rational numbers must be interspersed with the irrational numbers as points on the real line, each one following the other. There cannot be more irrational numbers than the rational numbers! 4 Where are all the Irrational Numbers? Almost all real numbers are supposed to be irrational. They should be pervasive but, after listing a handful like e, π, 2, and perhaps other square roots, it is hard to come up with more. Very few irrational numbers have been identified relative to the rational numbers, which begs the question: where are all the remaining irrational numbers? The integers are discrete points on the real line. Lots and lots and lots of rational numbers lie in the (relatively large) gaps between every two successive integers. There are many, many, many more rational numbers than integers but they have the same cardinality. Irrational numbers are hard to find vis-à-vis the rational numbers and, yet, they have higher cardinality. This is the exact opposite of what should be true. It is like the “upside down” parallel universe in the TV series “Stranger Things”. Irrational numbers may not exist (see Appendix E). Real numbers satisfy the Field axioms. Starting with the “initial” real numbers 0 and 1, the simple arithmetic in the Field axioms can only lead to rational numbers! How does the first irrational number get “created”? And how do the remaining come about in a way that they completely dominate all other types of numbers? Construction Models? Mathematics claims the following construction models yield the irrational numbers: ▪ Construction from Cauchy sequences ▪ Construction from Dedekind cuts ▪ Stevin’s construction This is just a theoretical claim of “construction” without a single example. There is no method outlined to calculate the irrational limit of a Cauchy sequence of rational numbers. There is no technique specified to compute a new Dedekind cut and, thereby, discover a previously unknown irrational number. There is no procedure prescribed to generate irrational numbers from Stevin’s infinite decimal representation. The logic seems to be - since irrational numbers exist, these methods yield irrational numbers - in theory even if they don’t in practice. 5 Dedekind Cuts Excerpts from the Wikipedia page “Dedekind cut” A Dedekind cut is а method of construction of the real numbers. It is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. Every real number, rational or not, is equated to one and only one cut of rationals. Mathematics is founded on the core principle of precise definitions (see Appendix A). Why is there only a “loose” characterization of the “gap”? The sequence {1/n} converges to the limit 0. Given any rational number x, the rational sequence {x - 1/n} converges to the rational limit x outside the sequence as all elements of the sequence are strictly less than x. Similar logic applies to the sequence {x + 1/n} from the other side of x.