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)]