Who discovered the difference between the infinities?

Question

As we know, there is a difference between the (infinite) size (or cardinality) of the integer numbers and the size of the reals ($\aleph_0$ and $\mathfrak c=2^{\aleph_0}$).

Who discovered it first?

Answer 1

Georg Cantor discovered it.

You can see at least : The Early Development of Set Theory :

in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the cardinality of $\mathbb R$ is strictly greater than that of $\mathbb N$.

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See :

• Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Engl.transl.by Philip Jourdain of "Beiträge zur Begründung der transfiniten Mengenlehre (I, 1895) and (II, 1897), 1915) , page 86,

for the definition of power or cardinal number of a set.

Answer 2

It will be rather difficult if not impossible to find the first one who discovered a difference between infinities. But it is clear that this one was not George Cantor (1845-1918). He came much later. (Cantor merely devised a certain, rather arbitrary, tool, namely the one-to-one correspondence or bijection, to base his theory upon it.)

A very old source is Robert Grosseteste (1168-1253) who said that actual infinite is definite. There are more moments in a long time interval than in a short one. The number of points in a segment one ell long is its true measure. https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf p. 106

Blaise Pascal (1623-1662) taught the existence of the three orders: Infinitely small, finite, and infinitely large (and applied it to body, mind, and God).

Gottfried Wilhelm Leibniz (1646-1716) also distinguished three grades of infinity.

Cantor himself mentions Bernard de Fontenelle (1657-1757) who invented actual infinite numbers. (G. Cantor, letter to A. Schmid, 26 March 1887, translated in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf p. 106)

We know that Leonhard Euler (1707-1783) accepted different infinities. $a/dx^2$ quantitas infinita infinities maior quam $a/dx$ (the first term is a quantity infinitely many infinities larger than the second). (W. Mückenheim: Die Geschichte des Unendlichen, 7. ed, Maro, Augsburg, p. 50)

Even way before Cantor Bernard Bolzano (1781-1848) distinguished infinities, for instance there are twice as many foci of ellipses than centres of ellipses. There are infinitely many more diameters of circles that centres of circles. (J. BERG (ed.): Bernard Bolzano, Wissenschaftslehre §§ 1-45, Friedrich Frommann Verlag, Stuttgart (1985), Bolzano-Gesamtausgabe, Reihe I Band 11,1, p. 31ff)

This is merely a short list, by no means complete, but sufficient to show that Cantor was not the first to distinguish different infinities.