WHY CANTOR’S THEOREM IS MEANINGLESS and there aren’t Infinite Infinities Pravin K. Johri Cantor’s theorem: The Power set of a set has higher cardinality than the set itself. Given an infinite set, one can recursively form Power sets of higher and higher cardinality leading to the conclusion there are infinitely many possible sizes of infinite sets. The notion of a single infinity is abstract. The very idea of “Infinite infinities” should boggle everyone’s mind. Excerpts from the Wikipedia page “Georg Cantor” Prior to this (Cantor’s) work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. It turns out that there is a hidden assumption in the axiom of Power set which is equivalent to the theorem’s conclusion. By assuming that Power sets exist via the axiom of Power set, Cantor’s theorem presumes the ultimate outcome and is meaningless for finite sets. Furthermore, it is not even defined for infinite sets! 1 The Two Notions of Infinity Aristotle’s abstract notion of a potential infinity is “something” without a bound and larger than any known number. 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. Few people realize that Cantor’s notion of a “completed infinity” is different than Aristotle’s idea of a potential infinity. It requires an undefined act of “completion” and a new abstract notation – the “aleph” numbers which represent the sizes of infinite sets. Cantor’s Theorem and Proof For any set S, the Power set of S denoted as P(S) has a strictly greater cardinality than the set S itself. When applied to an infinite set such as the countable set of natural numbers N, the Power set of N becomes uncountably infinite. The result is obvious for finite sets since the size of the Power set of a finite set with n elements is 2n. The proof for the infinite set N is by contradiction – assume there is a one-to-one correspondence between N and P(N) and find a contradiction. 2 Given any one-to-one correspondence, a natural number in the set N can be paired with a subset of N in P(N) that either contains or does not contain that natural number. If the former is true, the number is called “selfish” and called “unselfish” otherwise. The set of all unselfish numbers, denoted as D, provides the contradiction. This set D must be paired with a natural number, say d. By definition, d cannot be in D, but then it is an unselfish number and therefore must be in D, providing the contradiction. If D is empty then every natural number is selfish and must map to a set that contains the natural number, and in this case no natural number can map to the empty set in P(N) since the empty set contains no elements and, hence, includes no natural number. This proves that the cardinalities of the sets N and P(N) cannot be the same. Since P(N) contains N it has the larger uncountable cardinality (often |N| denoted as 1 = 2 where |N| = 0). Infinite Infinities The Power set of the Power set of N, and the Power set of that Power set, and so on indefinitely result in ever larger and larger sets. So, there can be infinite sizes of infinite sets denoted by the infinite sequence of aleph numbers 0, 1, 2 … Inconsistencies in the Proof of Cantor’s Theorem To show one-to-one correspondence, if it exists, between the set N and its Power set P(N) is difficult enough. To prove that there cannot be one-to-one correspondence between these two sets should be an order of magnitude harder! How was Cantor able to establish such a difficult proposition so concisely? Did the proof involve some artful, innovative argument? Not quite! Cantor’s proof relies on a number of questionable assumptions. Here is a list. 3 Why does the Power set P(N) contain the empty set while the set N does not? A set is defined as a fixed collection of distinct objects. S = {3, 5} is an example of a set that contains two different objects. A subset of a set includes one or more elements of that set and no other elements. The subsets of {3, 5} should be the two sets {3} and {5}. One could technically count the entire set {3, 5} as a subset of itself. The empty set denoted as {} is defined as the unique set with no elements. The empty set is inconsistent with both a set and a subset. If there is no element, there is no set (or a subset of another set). In fact, the empty set used to be called the null set which denotes that there is no set. The Power set of a set S, denoted as P(S), used to be the set of all subsets of S. This definition was later changed to include the empty set and the set S itself in addition to the subsets of S. With this revision, if S = {3, 5} then P(S) = {{}, {3}, {5}, {3, 5}}. Is the empty set forcibly included in the Power set to keep the proof of Cantor’s theorem whole? The Power set of the Power set of S, denoted as P(P(S)), includes the following objects: The empty set {} Subsets of size 1 {{}} {3} {5} {3, 5} Subsets of size 2 {{}, {3}} {{}, {5}} {{}, {3, 5}} {{3}, {5}} {{3}, {3, 5}} {{5}, {3, 5}} Subsets of size 3 {{}, {3}, {5}} {{}, {3}, {3, 5}} {{}, {5}, {3, 5}} {{3}, {5}, {3, 5}} The set P(S) {{}, {3}, {5}, {3, 5}} It has elements such as {} and {{}} which are considered to be distinct. The latter is just a set that contains the empty set. Can two “things” that both contain “nothing” be distinct? On the other hand, the empty set was characterized as unique, as one empty set is no different than another empty set. The concept becomes even more weird! 4 The Power sets can be formed recursively, without an end, eventually leading to infinite elements such as {}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, … All of which are a set that contains a set that contains a set that contains a set … that contains the empty set. Not a single element of the original set is present in any of these objects. Infinite distinct entities have been created out of nothing (the empty set). This theory is bizarre!! Cantor’s theorem breaks down if the empty set is not in P(N). The arbitrary inclusion of the empty set in P(N) ensures that the set D of all unselfish elements cannot be empty (or there is a contradiction with the one-to- one correspondence between N and P(N) assumed at the beginning of the proof). This is the key step that enables such a simple proof of Cantor’s theorem! If D is empty, then the remaining problem is still to show there is no one-to-one correspondence between N and P(N) – the same problem as before! The proof also breaks down if the empty set is included in the set N itself. It defies common sense that the addition of just one more element makes the size of a set increase to a higher order of infinity. Surely, if the theorem is indeed true, there must be a way to prove it without the controversial insertion of the empty set in the Power set. Does the Power Set of an Infinite Set Exist? Does P(N) exist? Mathematics does not provide an existence proof. Instead, it relies on the axiom of Power set which stipulates that the Power set exists for every set, finite or infinite. An axiom is a fancy name for an assumption. No infinite set (or quantity) can be constructed in reality. Simply assuming the existence of an infinite set and also that of the Power set of an infinite set are blatant violations of the concept of an axiom.