Cantor’s invented transfinite numbers
Page last updated 02 July 2021
Georg Cantor defended his invention of transfinite numbers as being a valid extension of finite numbers. In his own words: (Footnote: From Cantor, “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (The basis of a theory of sets) Leipzig 1883, as in p 182, Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts, ed Zermelo, Springer-Verlag, republished, 2013 )
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. In particular, when new numbers are introduced, it is only necessary to give definitions of them which will afford them a sufficient definiteness and, under certain circumstances, such a relationship to the previously established numbers that they can be clearly distinguished from one another in given cases. Once a number satisfies all these conditions, it can and must be considered as existing and real in mathematics
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
But the fact is that irrational numbers (both algebraic and transcendental), and also complex numbers, arose as solutions to equations and which have proven applications in a multitude of real world situations. Modern science and engineering would be impossible without them. By way of contrast, Cantor’s invented transfinite ‘numbers’ did not arise as an inevitable solution to an equation, and 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. )
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. Unfortunately, Cantor’s followers - instead of admitting that there was a fundamental problem with the theory, fell for the sunk cost fallacy, where one does not want to admit that the majority of the preceding time and effort are now of little value, instead insisting that the preceding time and effort still had real value.
And so, Cantor’s advice that only a theory which is free from contradictions should be permitted was ignored. And although no practical 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 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, a theory that 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”. 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.
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