"The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself when examining the categories of cardinal numbers and the order type, these basic notions of set theory, on the degree of their generality. [...] 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 (15 Feb 1896)]

"I have to draw your attention to two facts: 1st to the untearable ties that connect metaphysics and theology, in that the latter is the lodestar according to which the former is adjusting itself. [...] There follows a threefold: a) In metaphysical discussions it is sometimes inevitable to have theology join in. b) Every real progress in metaphysics strengthens or multiplies the tools of theology; it may even happen that human reason (of course subjugating to the infallible decisions of the church) may obtain deeper and richer symbolic insights with respect to the mysteries of the religion than has been expected or foreseen." [G. Cantor, letter to T. Esser (15 Feb 1896)]

"[...] it is clear that the theological considerations by which Cantor motivated his notion of the actual infinite, were metaphysical in nature." [A. Heyting: "Technique versus metaphysic in the calculus", in Imre Lakatos (ed.): "Problems in the philosophy of mathematics", North Holland, Amsterdam (1967) p. 43]

"Cantor is probably the last exponent of the Newtonian attitude with respect to religion." [H. Meschkowski, W. Nilson: "Georg Cantor Briefe", Springer, Berlin (1991) p. 15]

"[...] it was a certain satisfaction for me, how strange this may appear to you, to find in Exodus chapt. XV, verse 18 at least something reminiscent of transfinite numbers, namely the text: 'The Lord rules in infinity (eternity) and beyond.' I think this 'and beyond' hints to the fact that $\omega$ is not the end but that something is existing beyond." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"Compare the concurring perception of the whole sequence of numbers as an actually infinite quantum by St Augustin (De civitate Dei. lib. XII, chapt. 19) [...] While now St Augustin claims the total, intuitive perception of the set ($\nu$), 'quodam ineffabili modo', a parte Dei, he simultaneously acknowledges this set formally as an actual infinite entity, as a transfinitum, and we are forced to follow him in this matter." [G. Cantor, letter to A. Eulenburg (28 Feb 1886)]

So according to Cantor, the inventor of set theory, set theory $S$ by the following syllogism, is part of theology $T$:

$S\subseteq M \wedge M\subseteq T \Rightarrow S \subseteq T $

It appears as if today set theorists do not like to remember the roots. But have they ever been cut, and if so when and where?

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    $\begingroup$ Are you asking when the choice of axioms of set theory transitioned from what Cantor was convinced followed from his religious views to a shortlist mathematicians share? Probably around the time anyone other than him treated ST as axiomatic. $\endgroup$ – J.G. Apr 27 '18 at 12:26
  • $\begingroup$ @Alexandre Eremenko : Have you ever read Cantor's "Mitteilungen zur Lehre vom Transfiniten"? Do you know that the name cardinal numbers appears for the first time in correspondence with a Cardinal? $\endgroup$ – Wilhelm Apr 27 '18 at 13:44
  • $\begingroup$ @J.G. I do not ask about the axioms which were introduced by Zermelo in 1908 in his Grundlagen. I am asking whether the axioms were modelled on religious belief (like Euclid's geometric axioms were modelled on plane and space). In particular the existence of more real numbers than ever could be defined in the universe is a hint to God. $\endgroup$ – Wilhelm Apr 27 '18 at 14:08
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    $\begingroup$ According to available online etymology cardinal number was first used in the 1590's and so may be unrelated to Cantor's addressing a Cardinal. Latin cardinalis meaning "principal, chief, essential". $\endgroup$ – Nick R Apr 27 '18 at 17:39
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    $\begingroup$ Your claim that $M$ (metaphysics) is a subset of $T$ (theology) is, well, doubtful. Are you claiming that this was Cantor's belief? There may be connections as Cantor states, but proper inclusion does not follow. Certainly no philosopher would have made such a claim. $\endgroup$ – Nick R Apr 27 '18 at 18:13

It was never really connected. Cantor's mathematical papers contain nothing of the sort you cite. That Cantor himself was interested in theology is no more relevant than Newton's interest in the same. But you do not ask "when Mechanics threw of theology?" They were never related. But one person can write on several different subjects.

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    $\begingroup$ Newton did a lot of alchemy and theology, but did not base machanics or mathematics on it, contrary to Cantor. "One of the proofs starts from the notion of God and concludes first from the highest perfection of the Supreme Being on the possibility of the creation of a Transfinitum ordinatum, then from God's loving kindness and glory on the necessity of an actually created Transfinitum." (Cantor: Mitteilungen zur Lehre vom Transfiniten.) $\endgroup$ – Wilhelm Apr 27 '18 at 14:17
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    $\begingroup$ I was talking about his MATHEMATICAL papers. $\endgroup$ – Alexandre Eremenko Apr 27 '18 at 15:54
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    $\begingroup$ "Mitteilungen zur Lehre vom Transfiniten" is a mathematical paper. It contains an "extended, purely mathematical theory of order types". That is not only my opinion but a quote by Zermelo. It is simply impossible to distinguish set theory and theology (because set theory definitely belongs to metaphysics). $\endgroup$ – Wilhelm Apr 27 '18 at 16:19
  • $\begingroup$ "daß zu dem Begriffe einer Zahl die Endlichkeit derselben gehöre, und daß andrerseits das wahre Unendliche oder Absolute, welches Gott ist, keinerlei Determination gestattet. Was den letzteren Punkt anbetrifft, so stimme ich, wie es nicht anders sein kann, demselben völlig bei" ... "Omnia seu finita seu infinita definita sunt et excepto Deo ab intellectu determinari possunt."... " Thomas von Aquino, Opuscula, XLII de natura generis, cap. 19 et 20; XXXII de natura materiae et de dimensionibus interminatis." (Cantor: Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 1883) $\endgroup$ – Wilhelm Apr 27 '18 at 16:41
  • $\begingroup$ @Wilhelm: on my opinion, set theory belongs to mathematics, and Cantor discovered it when studying a concrete question about Fourier series. That Cantor himself connected it somehow to theology is of some historical interest, but this does not mean that set theory itself was related to theology. $\endgroup$ – Alexandre Eremenko Apr 28 '18 at 13:10

Cantor's metaphysical views are probably largely irrelevant compared to his lasting contribution to set-theoretic foundations commonly used today. Nevertheless there may be an interesting issue here having to do with mathematical modernity. Many historians today associate mathematical modernity with Cantorian set theory and Hilbert's formalism. On this view, Cantor is one of the early moderns. There is of course a tension here between Cantor's metaphysical views, on the one hand, and an attempt to portray him as a modern. Attitudes of this sort among historians become particularly problematic when they are combined with attempts to portray other mathematicians as countermodern and notably Felix Klein. Our assessment of the work of such historians, including Herbert Mehrtens and Jeremy Gray, is due to appear in 2017 in Mat.Stud. and is available on the arxiv.


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