Un-Real Analysis Why Mathematics is so Counterintuitive 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. In comparison, Calculus was a piece of cake. Now the professor seems to be speaking an entirely different language. This course starts with the development of finite set theory. A finite set is a very simple concept and easy to comprehend. Then, an infinite set is introduced and it is all downhill from this point onwards. This paper lays out many of the concepts in Real Analysis. Some are clear and intuitive, some are strange, and some are just bizarre. Finite Sets A finite set is defined as a fixed collection of distinct objects. The size or cardinality of a finite set is defined as the count of the objects in the set. There is a single noteworthy theorem to equate the cardinality of two sets if a bijection or one-to-one correspondence can be established between them. The best practical use of finite sets is in Venn diagrams. 1 Infinite Sets and the Two Notions of Infinity An infinite set is introduced as a set that is not finite. This definition is valid only if the theory developed so far included both finite and infinite sets. But it didn’t! So, what is an infinite set? Is it a finite set that is not finite? 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 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. The sets N and R are the infinite sets of natural numbers and of real numbers, respectively. Wade [9] is a standard undergraduate textbook. It starts by implicitly assuming R is a set. “We shall denote the set of real numbers by R.” Rudin [8] is a standard graduate textbook. It also assumes without reservation that R is a set. “A field is a set S with two operations, called addition and multiplication, which satisfy the so-called field axioms.” “There exists an ordered field R … The members of R are called real numbers.” Aristotle’s abstract notion of a potential infinity is “something” without a bound and larger than any known number. The sequence of finite natural numbers 1, 2, 3 … is potentially infinite. A different concept of a completed actual infinity is used with infinite sets. It requires a new abstract notation – the “aleph” numbers to represent the sizes of infinite sets 2 Excerpts from the Wikipedia page “Brouwer-Hilbert controversy” Cantor extended the intuitive notion of "the infinite" – one foot placed after the other in a never-ending march toward the horizon – to the notion of "a completed infinite" – the arrival "all the way, way out there" in one fell swoop, and he symbolized this [as] … ℵ0 (aleph-null). Excerpts from the Wikipedia page “Actual infinity” In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. What exactly is meant by “arrival all the way, way out there"? Why isn’t completion defined precisely as in other areas of mathematics (see Appendix A)? The completion of a metric space is obtained by adding the limits to the Cauchy sequences. The existence of infinite sets is an assumption in mathematics. Axiom of Infinity: There exists an infinite set N = {1, 2, 3 …}. The set N is a closed completed object but the sequence of natural numbers inside it is incomplete or endless. The natural numbers are all finite and yet the set N is actually infinite. The axiom of infinity conflates finite with nonfinite. Both the Peano and the Field axioms dictate that if numbers exist in the sets N and R then additional numbers exist as well. Neither the natural numbers nor the real numbers can be fully determined. There are always more numbers, and more, and yet more … Why must one “accept” N and R as completed objects? 3 A set is defined as a fixed collection of distinct objects. All elements of a set should be listed up front. Any rule or condition or notation involved in the formation of a set must be such that it can be fully exercised prior to the formation of the set, and all elements of the set determined exactly. As an example, the entity [1, 2, 3 … 10] can be fully expanded to yield the entire collection of objects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] which is a valid set. The Peano axioms and the Field axioms cannot be worked out to “completion”. They result in an unending list of numbers and are not a fixed collection of distinct objects. Just assuming that N is a set and formulating it as the axiom of infinity does not make it true! All kinds of infinite sets, some with rather exotic properties, have been conjured up in mathematics. The Cantor set and the “set of all sets that are not members of themselves” in Russell’s paradox to name a few. Can sets be based on contorted logic that cannot be fully resolved? Cardinality of Infinite Sets An infinite set has nonfinite cardinality. It is no longer the count of the objects in the set. Why is infinite cardinality defined differently? There was a prescription in the concept of actual infinity that the “size” of an infinite set is represented by the aleph number which is different than an ordinary number. The primary reason for the existence of ordinary numbers is to count objects and to measure quantities of physical entities. It is “irrational” that sets such as N and R are made up of ordinary numbers and, yet, their count cannot be a number! A new way has to be developed to compare the sizes of two infinite sets. 4 One-to-One Correspondence The notion of one-to-one correspondence or a bijection is carried over from finite sets to infinite sets. However, the one-to-one pairing is typically demonstrated only for an arbitrary finite element and not for all elements in the two sets, as in finite set theory. Let NE = {2, 4, 6 …} be the set of even natural numbers. It is a proper subset of N. The mapping n N → 2n NE is sufficient to show that these two sets have the same (infinite) cardinality. There is an implicit assumption that the natural numbers in the set N are fully enumerated by choosing an arbitrary finite n from it. Is this really true? The selection of a single finite n, no matter how large this n is, still leaves an unending stream of larger natural numbers (n+1, n+2, n+3 …) unaccounted for. Why doesn’t the Peano axiom (for every n, there is an n+1) figure in some way, shape or form (like an inductive argument) in how a bijection is established? If a method is improperly defined it will lead to inconsistent results and, sure enough, it does yield something that is not permissible in finite set theory. An infinite set can have the same cardinality as its proper subset! This is a contradiction as clear as it can be. And yet it is taken to be true and recharacterized as a counterintuitive result. The same definition of a set applies to both finite and infinite sets. Why do infinite sets have fundamentally different properties? The cardinality of infinite sets cannot be measured. Nevertheless, one can conclude two sets have the same cardinality. 5 The notions of one-to-one correspondence between infinite sets and the “same cardinality” are arbitrary at best and totally flawed at worst. Infinite sets seem to satisfy the “reverse” duck test which says that if it doesn’t look like a duck, doesn’t swim like a duck, and doesn’t quack like a duck, then it probably isn’t a duck. Two-to-One Correspondence In finite set theory, the notion of one-to-one correspondence can be easily generalized to the concept of two-to-one correspondence and even to m-to-n correspondence. Two-to-one correspondence exists between two sets A and B if every two elements in A are exactly paired with one element in B. As an example, if A = {1, 2, 3, 4} and B = {3, 4}, then the elements 1, 2 in A can be paired with the element 3 in B, and the elements 3, 4 in A can be paired with the element 4 in B. It can be easily proven that the set A has twice the cardinality as the set B. The mapping n-1, n N → n NE (n = 2, 4, 6 …) demonstrates two-to-one correspondence between N and NE and the cardinality of N should be twice that of NE.