Cantor’s invention of transfinite numbers
Page last updated 2 Sep 2021
Georg Cantor defended his invention of transfinite numbers as being a valid extension of finite numbers. In his own words: (Footnote: From Cantor, § 8 of “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (Foundations of a general theory of sets), Mathematische Annalen 21, Leipzig 1883, as in p 182, Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts, ed Zermelo, Springer-Verlag, republished, 2013 (my translation).)
every mathematical concept also carries the necessary corrective in itself; if it is sterile or inexpedient, it very soon shows it through its uselessness and it is then dropped because of lack of success.
the essence of mathematics lies precisely in its freedom
and: (Footnote: From Cantor, § 12 of “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (Foundations of a general theory of sets), Mathematische Annalen 21, Leipzig 1883, as in p 199, Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts, ed Zermelo, Springer-Verlag, republished, 2013 (my translation, see Grundlagen - English Translation).)
I therefore really do not know what should hold us back from this activity of forming new numbers, once it appears that the introduction of a new one of these innumerable number-classes has become desirable or even indispensable for the advancement of science.
But the fact is that irrational numbers (both algebraic and transcendental), and also complex numbers, arose as solutions to real number equations (Footnote:
For example complex numbers with integer real and integer imaginary parts can be referenced using only real numbers and simple equations, e.g. for the simultaneous equations:
x1 = x2 + 5
x2 · x2 = - 4
the solution for x2 is the square root of - 4 (the imaginary number 2i) and the solution for x1 is the complex number 5 + 2i. Furthermore, the concept of complex numbers arises directly from solutions to simple quadratic equations, depending on the coefficients, e.g. x2 - 4x + 8 = 0 has the solution whose roots are 2 + 2i and 2 - 2i. ) and which have proven applications in a multitude of real world situations, but there is no such equivalent derivation of transfinite numbers from the real numbers, which renders Cantor’s argument rather questionable. And modern science and engineering would be impossible without complex numbers. By way of contrast, since Cantor’s invented transfinite ‘numbers’ did not arise as an inevitable solution to any real number equation, it is no surprise that despite a babel-high tower of writings on these ‘numbers’ that has been in construction for over 130 years, there has not been a single useful real world application of these invented ‘numbers’. All real world science and engineering have progressed and continue to progress without any need for such invented numbers. (Footnote: Solomon Feferman: “The necessary use of higher set theory in mathematics of the finite has yet to be established. Furthermore, a case can be made that higher set theory is dispensable in scientifically applicable mathematics … Put in other terms: the actual infinite is not required for the mathematics of the physical world”, in ‘Infinity in Mathematics: Is Cantor Necessary?’, in Philosophical Topics 17.2 (1989): 23-45.) (Footnote: Despite no useful application ever having been found for transfinite numbers and the associated set theories, governments have still not realized that by supporting “research” into such matters they are simply throwing taxpayers’ money at people who are indulging in nothing more than a fantasy pastime which does not even have the redeeming quality of artistic merit.) For more on set theories, see the pages that give an overview of set theory, starting at Overview of set theory: Part 1: Different types of set theories.
At the time that Cantor wrote his apology (as previously quoted above) for his invented transfinite numbers, he was still developing his ideas of sets and these transfinite numbers, and he was absolutely convinced that he was on to something that was fundamentally eternally correct at its core, asserting: (Footnote: Georg Cantor, letter of 21st June 1888 to Carl Friedrich Heman, professor of theology at the University of Basel; as quoted in ‘Georg Cantor: His Mathematics and Philosophy of the Infinite’ by J W Dauben, Princeton University Press, 1990.)
“My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.”
The intensity of his conviction was due in large part to his religious convictions. He really believed that he was an instrument of god and was only revealing that which god had created, see Cantor’s religious beliefs and his transfinite numbers. Cantor brushed aside possible difficulties concerning these numbers by the convenient appeal that any such difficulties were surmountable by the infinite capability of god, even if not by the human mind.
However, after some further years of his work on his theories, Cantor began to realize (in the late 1890s) that there might be real problems inherent in his theory (besides the obvious contradiction of the notion of one infinity being bigger than another that he chose to allow on account of god’s infinite capability) such as the collection of contradictions that are generally known as Cantor’s paradox.
One version of Cantor’s paradox can be considered to be this: (Footnote: In fact, there are numerous ways of deriving a contradiction from Cantor’s set theory, this is only one such demonstration.)
By Cantor’s fundamental notion that sets can themselves be elements of sets, that means that there can be a universal set U of all sets (i.e: every set is an element of U).
But it is also the case that for every set A there is the power set P(A) of that set, which is the set of all subsets of the set A (i.e: every subset is an element of the power set P(A)).
Here is the paradox:
If the universal set U is the set of all sets, then its power set P(U) must be a subset of U, and so, by Cantor’s set theory, it must be either the “same size” or “smaller” than U, and this applies also to infinitely large sets. (Footnote: Note that the power set must also be an element of U.)
But in Cantor’s set theory, every power set is “bigger” than the set from which it is derived (has a bigger cardinal number), and this applies also to infinitely large sets.
So the cardinal number of the universal set U must be bigger than itself - a contradiction.
This paradox and the subsequent discovery of an even bigger problem - Russell’s paradox - shook Cantor to very essence of his being and he never recovered from these findings, spending most of the rest of his life in sanatoria. Gerhard Kowalewski (1876 - 1950), who had quite regular contact with Cantor in the 1890’s, remarked that: (Footnote: Gerhard Kowalewski, Bestand und Wandel. Oldenbourg Wissenschaftsverlag, Munich 1950, pp. 106-107, my translation. The “by way of contrast” refers to the contrast between Cantor’s underlying philosophy and that of Leopold Kronecker, which was the subject of the previous sentences.)
“Cantor, by way of contrast, set up the principle: ‘The essence of mathematics lies in its freedom.’ He did not want to have his wings tied. As he continued to develop his theory, he was destined to encounter the paradoxes which emerged by allowing, as Cantor did, too great a degree of freedom to mathematical thinking.”
Unfortunately, Cantor’s followers - instead of admitting that there was a fundamental problem with the theory, fell for the sunk cost fallacy, where rather than admit that one has taken a wrong turn that leads to a dead end and that the majority of the preceding time and effort are now of little value, one insists that the preceding time and effort still has real value and possibilities for future progression.
And so, Cantor’s advice that only a theory which is free from contradictions should be permitted was ignored. And although no practical or scientific utility had been found for transfinite numbers, Cantor’s recommendation that a mathematical theory that is useless should be dropped was not observed. (Footnote: And today, after over 130 years of transfinite theory, there is still no useful practical or scientific application of transfinite numbers.)
Instead, after the discovery of Russell’s paradox, Cantor’s followers continued for the next 25 years or so to throw time and effort into patching up Cantor’s theory, instead of starting over from fundamental logical principles. The result was Zermelo-Fraenkel set theory, a theory which introduced ad hoc axioms to patch up the defects, and which never really satisfied Zermelo himself, (Footnote:
Gerhard Kowalewski (1876 - 1950), a mathematician of the same era and well acquainted with both and Zermelo and Cantor, wrote that Zermelo was never fully satisfied with his version of set theory:
Since Zermelo gave a proof of the well-ordering theorem, we have known that there are no other cardinal numbers than the Alephs. With this an undreamt-of deep insight into the essence of the infinite is gained, almost too deep for the human spirit. No wonder that this bold flow of thought leads to paradoxes. The question of which restrictions must be introduced in order to remove the paradoxes is difficult to resolve. Zermelo’s attempt in this direction remained flawed, as the great researcher himself admits.
From the autobiography of Gerhard Kowalewski, Bestand und Wandel, Oldenbourg Wissenschaftsverlag, Munich 1950, p. 250, my translation.) but which its proponents claim is the one true mathematical foundation, despite its many varieties (Footnote: The standard set theory used today is called Zermelo-Fraenkel set theory, after Ernst Zermelo and Abraham Fraenkel, who devised the theory in the early 20th century. It comes in several varieties, which you might find surprising when the proponents of set theory claim that set theory is the one true fundamental theory of mathematics. One variety includes what is called the Axiom of Choice, which essentially assumes that even if one cannot define a set whose elements are taken from infinitely many sets, don’t worry - there exists a metaphysical entity that will do it for you, and so, such a set “exists” (see also The Axiom of Choice). Another variety of set theory asserts that the Continuum Hypothesis is true. So you can have a set theory where the Axiom of Choice applies and where the Continuum Hypothesis is true, and a set theory where the Axiom of Choice does not apply and where the Continuum Hypothesis is true, and so on, and so on … . Also note that there are several other similar set theories have been proposed as being better than Zermelo-Fraenkel theory.) and despite the bastard nature of its turbulent birth. For more on this, see the pages that give an overview of set theory, starting at Overview of set theory: Part 1: Different types of set theories.
You can read online an English translation of one of Cantor’s major works, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets), which lays out his philosophy on different sizes of infinity and transfinite numbers. The page The Origins of Transfinite Numbers gives the details of the process by which Cantor arrived at his notion of transfinite ordinal numbers, and the page Cardinal numbers discusses in detail Cantor’s transfinite cardinal numbers.
Cantor’s “Freedom of Mathematics”
It is richly ironic that today’s conventional mathematics has completely forgotten the roots that gave rise to it. At the time that Cantor was trying to get his ideas accepted in the face of opposition from people like Leopold Kronecker, he became a champion for the idea that there should be complete freedom in mathematics, provided that it satisfies certain conditions such as preventing contradictions. He says in Section 8 of his Grundlagen that:
“Mathematics is completely free in its development and is only bound to the self-evident consideration that its concepts are both free of contradictions and that they are in fixed relationships to proven concepts that have already been previously established … every mathematical concept also carries the necessary corrective in itself; if it is sterile or inexpedient, it very soon shows it through its uselessness and it is then dropped because of lack of success. On the other hand, every superfluous constriction of the impulse for mathematical research seems to me to bring with it a much greater danger, and one that is all the greater as no justification can really be drawn for it from the nature of science; for the essence of mathematics lies precisely in its freedom.”
However since that time, Cantor’s pleas for freedom in mathematics have become corrupted beyond recognition. Today it is impossible to get published in any mainstream mathematical journal any criticism of today’s set theories whose origins can be traced back directly to Cantor’s notion of freedom of mathematics. It is forbidden to suggest that the contradictions that arise in such theories indicate that there is a problem with such theories; in fact, one is not even allowed to call them contradictions, the only allowable term is “paradox”, fostering the pretense that giving something a different name can somehow change its intrinsic character.
The parallel to similar situations in politics is striking - many political movements begin with a clamor for freedom, but all too often it leads to an autocracy where its particular brand of “freedom” becomes the only established permitted “freedom”, and any notions of a freedom that is not in accordance with its rules is strictly policed and prohibited.
And the result, as in such political autocracies, is the stifling of new ideas, the cessation of progress, and a ubiquitous dull monotonous mediocracy whose primary objective is maintaining the status quo.