Cantor devised set theory for application to reality but started from the divine wisdom of the hierarchy of infinities. My question is: What of Cantor's claims has become reality? What are practical applications of set theory? And what parts of transfinite set theory require the philosophical position of platonism?

Here are some quotes which may help to familiarize with the stuff.

In a letter to Hilbert he wrote about his plan of a paper on set theory and its applications: "The third part contains the applications of set theory to the natural sciences: physics, chemistry, mineralogy, botany, zoology, anthropology, biology, physiology, medicine etc. It is what Englishmen call 'natural philosophy'. In addition we have the so called 'humanities', which, in my opinion, have to be called natural sciences too, because also the 'mind' belongs to nature." [G. Cantor, letter to D. Hilbert (20 Sept 1912)]

Cantor explained his impetus for devising set theory to Mittag-Leffler: "Further I am busy with scrutinizing the applications of set theory to the physiology of organisms. [...] I have been occupied for 14 years with these ideas of a closer exploration of the basic nature of all organic; they are the true reason why I have undertaken the painstaking and hardly rewarding business of investigating point sets, and all the time never lost sight of it, not for a moment. Further I am interested, purely theoretically, in the nature of the states and what belongs to them, because I have my opinions on that topic which later may become formulated mathematically; the striking impression that you perhaps may obtain will disappear, when you consider that also the state in some sense represents an organic being." [G. Cantor, letter to G. Mittag-Leffler (22 Sept 1884)]

"By applied set theory I understand what usually is called physical science or cosmology. To this realm all so-called natural sciences are belonging, those concerning the anorganic as well as the organic world. [...] For mathematical physics the theory of types is particularly important because the latter theory is a powerful and sharp tool for the discovery and the intellectual construction of the so-called matter. Related to this is the applicability of the theory of types in chemistry. [...] Of very special interest seems to me the application of mathematical type theory on study and research in the realm of the organic." [G. Cantor, letter to G. Mittag-Leffler (18 Nov 1884)]

"This has created my desire to replace the mechanical explanation of nature by a more complete one, which I would call in opposition to the former an 'organic' one." [G. Cantor, letter to W. Wundt (4 March 1883)]

"The actual infinite in abstracto and in concreto, however, where I call it transfinitum, are not only subject of an extended number theory but also, as I hope to show, of an advanced natural science and physics." [G. Cantor, letter to I. Carbonnelle (28 Nov 1885)]

Cantor says that he has no safer knowledge of anything in nature than of his transfinite set theory. "Therefore I am convinced that this theory one day will belong to the common property of objective science". [G. Cantor, letter to I. Jeiler (20 May 1888, Whitsun)]

Finally Cantor devised transfinite set theory in order to defend Christian religion. "The time is not far, however, that my teaching will turn out to be a really exterminating weapon against all pantheism, positivism and materialism." [G. Cantor, letter to J. Hontheim (21 Dec 1893)]

"The general set theory [...] definitely belongs to metaphysics. [...] and the fact that my presently written work is issued in mathematical journals does not modify the metaphysical contents and character of this work. [...] By me Christian philosophy is for the first time confronted with the true teachings of the infinite in its beginnings. [G. Cantor, letter to T. Esser (1/15 Feb 1896)]

Concluding: "If one has recognized the truth of something, then one knows to be in possession of the truth and one feels [...] sort of duty, as far and as long as power reaches, to tell it to others." [G. Cantor, draft of a letter to A. Schmid (18 April 1887)]

closed as unclear what you're asking by Conifold, Joseph Weissman Aug 26 '17 at 22:44

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    The only common positions in philosophy of mathematics that would reject transfinite set theory would be constructivism and finitism (some nominalists do as well). You don't need platonism to have infinite sets: take for example Hilbert's formalism, as long as the symbols all obey the rules set out it doesn't matter if sets are real abstract objects or not, if it works it works. Axiomatic set theory is perfectly fine under a formalist view, it then just becomes the case that we're talking about symbols and rules for symbol manipulation, not actual infinite objects. – Not_Here Aug 20 '17 at 17:24
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    Set theory is a mathematical discipline today and has been since the '60s, it's no longer something that only philosophers and logicians talk about. So, it is interesting to realize that a lot of (not all) working set theorists don't particularly care about the philosophy of mathematics, in the same way that someone who studies algebraic geometry doesn't care if rings are really abstract objects or not. Yes, historically set theory is connected to philosophy of mathematics, but contemporary working set theory isn't philosophical, its mathematical so set theory doesnt require a position to work – Not_Here Aug 20 '17 at 17:30
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    You have three different questions in the first paragraph, our policy is one question per question. Moreover, the first two are unrelated to the title question, and so is the rest of the post. The title question is unclear because "accept" is ambiguous (as mathematical practice, as symbolic game, as metaphysics?), and disambiguating it will pretty much answer the question. So it is unclear what you are asking. – Conifold Aug 22 '17 at 20:28
  • @Conifold: The questions are not as different as they might appear. Set theory was devised by Cantor for practial application in science etc. If set theory has been successful in some domain, then it has acquired reality, and no platonism is necessary to accept it. I am asking for such cases. In other fields, where it was not successful, Platonism would be required. – Heinrich Aug 23 '17 at 13:23
  • @Heinrich No it wouldn't, most schools of mathematical anti-realism agree that mathematical statements have truth values, which means they would agree to use transfinite set theory as a foundation for mathematics, but they still reject platonism and that mathematical objects are real. Just look at the example of Hilbert's formalism, there is nothing about formalism that requires platonism or for the theory to be realized in a scientific field, they just require that the game of symbol manipulation works, which is does, so zero platonism is required for them. – Not_Here Aug 23 '17 at 14:51

Whitehead and Russell in Principia Mathematica showed that Cantor's notion of one-to-one correspondence is sufficient to deduce transfinite number:

  • A transfinite number is defined as the cardinal of a reflexive class;
  • A reflexive class is any class that has a one-one relation to a proper part of itself;
  • A proper part is a part not the whole.

We can see that there is no platonism required.

  • Your second point defines Dedekind-infinity. That is the same as potential infinity or simply the infinite applied in analysis. Whether we call it oo or omega or aleph_0 and classify it as a transfinite "number" does not matter in mathematics. For the acceptance of uncountable sets with their necessarily undefinable elements Platonism is required. Otherwise most elements of mathematics are not existing as individuals. That makes mathematics resemble a religion with its Gods and angels (which however exist at least as individuals having names and can be addressed individually). – Heinrich Aug 22 '17 at 12:07
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    @Heinrich You do realize that defining all objects is as impossible as proving all statements. The sequence of natural numbers is neither "ascertained" nor "guaranteed" by axioms of arithmetic, or by anything else. At most, it is motivated by experience with (relatively small) integers. And if mathematics "does not concern external world" why should axioms of set theory be treated any differently than of arithmetic or geometry? – Conifold Aug 23 '17 at 0:11
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    @Heinrich - you are mistaking a definition for implication. A definition in PM is typographic convenience. "A is defined as such and such" does not mean "A implies such and such." This false equivalence has been considered as trivial by modern mathematicians and has been major source of confusion. – George Chen Aug 23 '17 at 16:50
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    @Georg Chen: A class that does not exist means that class is empty. A class that is empty means that class does not exist. That is not a false equivalence. It simply is an equivalence. My definition of to exist: Everything exists that can be defined such that two educated mathematicians can talk about it understanding the same. So the empty set as well as the set of real numbers exists, but most real numbers do not. Sorry, I will not continue this interesting discussion because of the moderator's warning. – Heinrich Aug 23 '17 at 19:52
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    @Heinrich - You are right: that definition does amount to equivalence, and transposition does apply(I was wrong in the last comment). ✳24.53 does say empty class does not exist, but in PM, none-exist is just another way of saying empty. We definitely have different understandings of what exist means here. I admit that before your challenge my understanding of this chapter was shallow. – George Chen Aug 23 '17 at 20:39

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