The Flaw in Mathematics Mistakes made in Infinite Set Theory more than a Century ago Pravin K. Johri Cantor’s set theory in the form of the axiomatic Zermelo-Fraenkel (ZFC) set theory is used in all areas of mathematics as a foundational basis for proving theorems. The object N = {1, 2, 3 …} is taken to be a completed, fixed, infinite set. It is really just an assumption, formulated as the axiom of infinity, and is simply based on faith that completed infinite entities exist. A fundamental mistake has already been made! The elements of N, the natural numbers, are all required to be finite and yet the size of N is taken to be an actual infinite quantity denoted as ℵ0 (aleph- null). Finite logic, which manipulates finite numbers, is used to prove properties of infinite objects without any specifics on how this finite logic extends to the actual infinite. This conflates finite with infinite. It is not clear what ℵ0 is, other than it is not a number, it is not finite, and it cannot be measured. A pure science, which should be concerned with numbers only, is now dabbling in ill-defined non-numbers. No wonder the rest of set theory is full of bizarre results, logic that defies common sense and is applied selectively to prove some results but not others, and many outright contradictions. 1 Natural and Real Numbers There is an endless sequence of finite natural numbers 1, 2, 3 … based on the Peano axioms that 1 is a natural number, and if n is a natural number then so is n+1 for all n. 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. Both the Peano and the Field axioms dictate that if numbers exist 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 … Finite Sets A finite set is defined as a fixed collection of distinct objects. The size of a finite set is defined as the count of the objects in the set. A fancy term, cardinality, is used to denote size because, later on, some infinite sets will turn out to be uncountable. It is easy to construct mappings between sets as the elements of a finite set are indexed. Mappings are characterized as injective, surjective, and bijective which are all intuitive concepts. There is a single noteworthy theorem to equate the cardinality of two sets if one-to- one correspondence or a bijection can be established between them. A bijection is not necessary to show that two finite sets have the same size. One can simply count the objects in the two sets. It will be required with infinite sets. The best practical use of finite sets is in Venn diagrams. 2 Infinite Sets The same definition of a set as a fixed collection of distinct objects continues to apply. 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.” The two Notions of Infinity 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 Excerpt 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). Excerpt 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. 3 Why must one “accept” N and R as completed infinite objects? What exactly is meant by “arrival all the way, way out there"? How can one “arrive” at ℵ0 if the natural numbers do not increase to ℵ0? Why isn’t completion defined precisely as in other areas of mathematics? The completion of a metric space is obtained by adding the limits to the Cauchy sequences. Can completion occur without a single modification to the potentially infinite, incomplete sequence 1, 2, 3 … inside the set N? Modern Mathematics Hilbert’s formalism or axiomatic approach is used in modern mathematics and results are established based on the core concepts: Precise definitions Logically correct arguments According to Quinn this “non-scientific approach” provides “unexpected bonuses”. Excerpts from Quinn [6] The breakthrough (in mathematics) was development of a system of rules and procedures that really worked, in the sense that, if they are followed very carefully, then arguments without rule violations give completely reliable conclusions. It became possible, for instance, to see that some intuitively outrageous things are nonetheless true. The definitions include the axioms. An axiom is a fancy word for a starting assumption which is taken to be valid and is not proven. In the past axioms had to be intuitive and also self- evident, but this is no longer required in modern mathematics. Axioms can be arbitrary in Hilbert’s formalism and they have to be stated up front. However, Cantor’s original “naïve” set theory resulted in paradoxes and new axioms were added in the “non-naïve” ZFC to rule out these anomalies, a clear violation of the concept of an axiom! The existence of infinite sets is cast as an axiom in modern mathematics. This is unfortunate. Axioms are not meant to institute as a fait accompli a proposition which otherwise is hard to establish and may even be wrong. 4 Axiom of Infinity There exists at least one infinite set namely the set N = {1, 2, 3 …}. The axiom of infinity is neither intuitive nor is it self-evident. In fact, it contradicts itself. The set N is considered a completed object even though the sequence of natural numbers inside it is incomplete. The Peano axioms cannot be resolved fully. They result in an endless non-fixed list of numbers which cannot be a fixed collection of distinct objects. Here is how Fraenkel describes the “actual” infinite set N = {1, 2, 3 …}. Excerpts from the Wikipedia page “Actual infinity” If the positive number n becomes infinitely great, the expression 1/n goes to naught (or gets infinitely small). In this sense one speaks of the improper or potential infinite. In sharp and clear contrast the set just considered is a readily finished, locked infinite set, fixed in itself, containing infinitely many exactly defined elements (the natural numbers) none more and none less. The set N is neither readily finished nor is it fixed in itself. There is no sharp and clear contrast, just a glaring self-contradiction. The natural numbers are all finite and yet the set of natural numbers is actually infinite. The natural numbers are called the counting numbers and they steadily increase in steps of 1. If there are infinitely many exactly defined elements then this cannot be true without there being an infinite counting number. But there isn’t one! The axiom of infinity conflates finite with nonfinite. This conflation is the reason behind the many “intuitively outrageous things” or contradictions. It allows finite logic, which doesn’t complete, to be used with infinite objects. Just assuming that N is a set and formulating it as the axiom of infinity does not make it true! 5 Cardinality of Infinite Sets An infinite set has nonfinite cardinality. It is no longer the count of the objects in the set. The same definition of a set applies to both finite and infinite sets. So, why is the size of an infinite set defined differently? There is a prescription in the concept of actual infinity that “infinity” is represented by an aleph (non)number, a bizarre concept. No one really knows what it means other than an Aleph number is not an ordinary number. It is confusing and wrong to call it a “number”. Ordinary numbers are used to count objects. It is “irrational” that sets such as N and R are made up of ordinary numbers and, yet, their count of elements cannot be such a number and, instead, has to be a non-number! A new way is required to compare the sizes of two infinite sets.