Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back to ancient era.
Question: Was there anybody (mathematician, philosopher, etc.) before Cantor who mentioned (or conjectured) existence of infinities of different sizes even if he stated this claim informally? Please introduce your references.
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See Galileo's Paradox with the English transaltion of the relevant part of
Simpl. Qui nasce subito il dubbio, che mi pare insolubile: ed è, che sendo noi sicuri trovarsi linee una maggior dell'altra, tutta volta che amendue contenghino punti infiniti, bisogna confessare trovarsi nel medesimo genere una cosa maggior dell'infinito, perché la infinità de i punti della linea maggiore eccederà l'infinità de i punti della minore. Ora questo darsi un infinito maggior dell'infinito mi par concetto da non poter esser capito in verun modo.
Salv. Queste son di quelle difficoltà che derivano dal discorrer che noi facciamo col nostro intelletto finito intorno a gl'infiniti, dandogli quelli attributi che noi diamo alle cose finite e terminate; il che penso che sia inconveniente, perché stimo che questi attributi di maggioranza, minorità ed egualità non convenghino a gl'infiniti, de i quali non si può dire, uno esser maggiore o minore o eguale all'altro. Per prova di che già mi sovvenne un sì fatto discorso, il quale per più chiara esplicazione proporrò per interrogazioni al Sig. Simplicio, che ha mossa la difficoltà.
Here the English translation [from Wikipedia : Galileo Galilei, Dialogues concerning two new sciences, transl. Crew and de Salvio (Dover reprint, 1954), pp.31–33] :
Simplicio : Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity [emphasis added], because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.
Salviati : This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.
[...]
Sagredo : What then must one conclude under these circumstances?
Salviati : So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.
So Galileo's conclusion was not a pre-cantorian theory of infinites of different size but the clear understanding that some "properties" of finite "magnitudes" (like : "equal", "greater" and "less") are not applicable to the infinite ones.
This discussion about the Paradox of infinite traces back to medieval philosophy; see Nicole Oresme using
the principle of one-to-one correspondence [to show] that the collection of odd natural numbers is not smaller than the collection of natural numbers, because it is possible to count the odd natural numbers by the natural numbers [para 2.6 Mathematics, and also the reference to Bearwardine].
Oresme shows that of two actual infinites neither is greater or smaller than the other. [...] Oresme's result does not necessarily imply equality between actual infinites. Moreover Oresme shows that cases can be conceived in which two infinites can be regarded as unequal, but this unequality is not to be understood in the sense of ‘smaller’ or ‘greater’ (Oresme does not contradict himself), but rather in the sense of ‘different’.
Since comparable quantities are either equal to one another or one is smaller or greater than the other, Oresme concludes that actual infinites are incomparable: that is, that notions like ‘smaller’, ‘bigger’, and ‘equal’ do not apply to infinites.
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According to this reference in 400BC, Jain mathematicians distinguished between five types of infinity : infinite in one direction, infinite in two directions, infinite in area, infinite everywhere, and infinite perpetually.
Of course, this is not entirely a proper distinction, however one could say that the perpetually infinite is distinct from the spacially infinite, so they may have correctly identified two different infinities.
Regarding Cantor, his well-documented comment - "I see it but I don't believe it" - would seem to suggest that he believed he was the first to explode these ideas (formally).
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The assertion that Cantor was the first to deal with infinities of different sizes is a bit misleading. This may be true in the context of set theory that he introduced, but there are different ways of formalizing infinity. Two centuries before Cantor, Leibniz dealt with a hierarchy of infinite numbers (inverses of his hierarchy of infinitesimals $dx$, $dx^2$, etc) in his treatment of infinitesimal calculus. One of the works dealing with this is his text Cum Prodiisset of 1701; another is his text on the transcendental law of homogeneity dating from 1710. This and related issues are discussed in this article in the Notices AMS.
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Bernard Bolzano (1781-1848) distinguished different infinities, for instance there are twice as many foci of ellipses than centres of ellipses. There are infinitely many more diameters of circles than centres of circles. J. Berg (ed.): Bernard Bolzano, Wissenschaftslehre §§ 1-45, Friedrich Frommann Verlag, Stuttgart (1985), Bolzano-Gesamtausgabe, Series I Vol. 11,1, pp. 31f
Bolzano did not accept infinite bijections as a means to measure infinities but applied reasoning based on analysis like the following: For every finite initial segment of natural numbers there are about 50 % even numbers. This holds the better the larger the the segment is. Hence it holds in the limit too.
In the ancient India, Mahavira, the founder of Jainism became very close to the modern notion of "different sizes of infinities". In the mathematical part of Jain philosophy, there are several kinds of infinities.
Quoted from Wikipedia:
For further explanations see also this link.