The Cantor Set A Perfect, Uncountable Set with no Segments or A Perfect Example of all that is Wrong with Cantor’s Infinite Set Theory Pravin K. Johri The first course in Real Analysis or Analysis is shocking to most students. Mathematics is suddenly counterintuitive and so difficult to comprehend. The course steadily becomes more and more bizarre. Then, towards the end, the student encounters the Cantor set. So far, infinite sets existed and had nonintuitive properties. Now, one can be constructed using an infinite, never- ending procedure of subdividing the unit interval. It involves infinite sequences which previously could not reach their limits but now do, somehow. Mathematics does not specify what changed to allow such a radically different outcome. The subintervals have some characteristics the Cantor set inherits and some it doesn’t without a clear explanation why. There are other unreal properties which are often assigned as homework and one can waste entire weekends trying to make sense of it all. It is an apt culmination of the course. The Cantor set embodies everything that is wrong with Cantor’s infinite set theory starting from the initial assumptions of an actual infinity and of infinite sets. Both are simply accepted to be true in mathematics, a pure science, and cannot be disproven as there is no proof to start. The only way to show that the theory is totally unfounded is to create one inconsistency after another. And Cantor’s theory is full of contradictions. There is an entire Wikipedia page devoted to it. 1 The Cantor Set Consider subsets of the unit interval [0, 1]: A1 = [0, 1/3] U [2/3, 1] A2 = [0, 1/9] U [2/9, 3/9] U [6/9, 7/9] U [8/9, 1] And so on. ∞ Then C = ⋂푛=1 퐴푛 is the ternary Cantor set. n -n An is made up of 2 intervals each of length 3 and yet the Cantor set contains no segments. The first step removes a segment of length 1/3, the second a segment of length 2/9, and so on. The total length removed is 1/3 + 2/9 + 4/27 + … = 1. The entire length of the unit interval is excised and yet the Cantor set contains uncountable points. Excerpts from the Wikipedia page “Cantor Set” However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points. It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third of that, and in the top third of that, and so on ad infinitum - alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. The number 3/10 is also in the Cantor set and is not an endpoint. … there are as many points in the Cantor set as there are in [0, 1], and the Cantor set is uncountable … However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. 2 Excerpts from the Wikipedia page “Cantor Set” To see this, we show that there is a function f from the Cantor set C to the closed interval [0, 1] that is surjective (i.e. f maps from C onto [0, 1]) so that the cardinality of C is no less than that of [0, 1]. … For a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s. … The function f from C to [0, 1] is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number. The Cantor set is also a perfect set in that no point in it is an isolated point. These properties are truly unbelievable - in that they should not be believed. Ironically, the Cantor set is a perfect illustration of everything that is wrong in Cantor’s infinite set theory. Inconsistency 1: Is this a valid procedure? The first issue is whether the Cantor set can be formed in the way it is described. The sequence of natural numbers 1, 2, 3 … is endless. The process of forming the subsets An should go on forever. But it terminates and produces a set without a clear description of how the ending comes about. For segments to disappear the length of the segments, 3-n, must shrink to zero. At the same time, there is no minimum value of a real number and the number 1 cannot be reduced to 0 by repeatedly dividing it by 3. A clear, precise description is required of what happens when sequences are used within sets to diametrically change their behavior. The way numbers are defined there is an undefined gap or middle (see Appendix D). Some infinite procedures in mathematics are allowed to jump the gap, others are not. No procedure in a pure science should be based on simply the “acceptance” that completion occurs (see Appendix B), and that it can be arbitrarily applied to some situations but not to others. 3 Inconsistency 2: The undefined ultimate step (completion) The Cantor set procedure creates sequences of nested closed intervals. In = [an, bn], n N, are nested closed intervals if I1 I2 I3 … Nested Closed Interval Theorem: ∞ If |In| → 0 as n → , then T = ⋂푛=1 I푛 contains exactly one number. The sequence of nested intervals is characterized by two sequences {an} and {bn} with an bn for all n N. Hence, an a b bn and T = [a, b]. Further, T = [a, b] = a = b if |In| → 0. This theorem violates the Field axioms! A positive quantity cannot ever reduce to zero through division. There is no minimum value of a real number, a direct consequence of the field axiom mandating a unique multiplicative inverse, and 3-n cannot shrink to 0 no matter how large n becomes. For example, the first interval in each step of the Cantor set procedure has the form [0, 3-n] which becomes progressively smaller as n increases. Each and every one of these nested intervals has positive length and contains uncountable real numbers. Yet, somehow, the ∞ −푛 infinite intersection ⋂푛=1[0, 3 ] of the very same intervals no longer has this property. If the non-degenerate nested intervals [0, 3-n] shrink to a single point then there must be some, as yet, undefined ultimate step that takes uncountable points (a higher order of infinity than ℵ0) down to a single point at one fell swoop. This concept is called completion of infinite entities and mathematicians just accept that it happens (see Appendix B) for some entities (intersection of nested intervals) but not others (sequences such as {3-n}). If such an important notion is just based on faith then mathematics is no different than a religion. The onus is on mathematics to precisely define how it happens, in accordance with its core principles (see Appendix A) and remove the inconsistency in how it is applied. 4 Inconsistency 3: The use of finite logic The conclusions regarding the Cantor set rely on finite logic. Many claims are made that are clearly true when the index n is finite. It is claimed that the endpoints of all remaining intervals in any step of the Cantor set procedure are left behind in the Cantor set because they are not excised in any finite step of the procedure. But these endpoints are all rational numbers and are countable. An additional claim is made that interior points of intervals are also left behind because interior points such as 1/4 and 3/10 are not removed in a finite number of steps. Almost all members of the Cantor set must be interior points and also irrational numbers, or it cannot be uncountable. In direct contrast, every step has intervals but none remain at the end. The sequences characterizing the intervals and their limits need not have consistent properties. The sequence {3-n} has all positive terms but its limit 0 is not positive. Finite logic does not necessarily extend to the limiting values! A convergent infinite sequence like {1/n} can come arbitrarily close to its limit 0 but not quite reach it. However, the infinite intersection of the nested closed intervals defined by the same ∞ sequence and its negative equivalent, ⋂푛=1 [-1/n, 1/n] reaches the single limiting value 0, which is an interior point of all intervals in the procedure.