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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?

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    $\begingroup$ Georg Cantor. $\endgroup$ – Mauro ALLEGRANZA Feb 14 '15 at 17:11
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    $\begingroup$ @Mauro: Why not make it an answer, so peterh can accept it, and it will be here for future users? $\endgroup$ – Gerald Edgar Feb 14 '15 at 17:48
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    $\begingroup$ Please don't use notation incorrectly: $\aleph_1$ is not the notation for the "infinity of the reals". $\endgroup$ – Andrés E. Caicedo Feb 14 '15 at 19:12
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    $\begingroup$ @AndresCaicedo Thank you the correction - sorry, I am a programmer :-) $\endgroup$ – peterh says reinstate Monica Feb 14 '15 at 19:19
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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 $\mathbb N$.


See :

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

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    $\begingroup$ There is a bit more to add, namely that the very definition of cardinality is due to Cantor. $\endgroup$ – Andrés E. Caicedo Feb 14 '15 at 19:15
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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.

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  • $\begingroup$ This was a very interesting answer. I wonder why it got downvoted. $\endgroup$ – DukeZhou Jun 25 '17 at 22:06
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    $\begingroup$ @DukeZhou: The reason is simply the sentence "Cantor merely devised a certain, rather arbitrary, tool". There are many people who adore Cantor so much that they consider this sentence a sacrilege. If someone even dares to say that this is not only an arbitrary but a useless tool because "for every n in N: n belongs to a finite initial segment which is followed by infinitely many natural numbers such that general quantification fails for infinite sets and equicardinality does not prove anything about same number of elements", the statemenet will be heavily downvotes or even deleted. $\endgroup$ – Otto Jun 26 '17 at 11:12
  • $\begingroup$ I did not downvote, but I can understand the downvote: the question asks explicitly about the discovery that naturals and reals have different cardinalities. I don't see how your answer addresses that. $\endgroup$ – Michael Bächtold Jun 26 '17 at 14:43
  • $\begingroup$ @Michael Bächtold: The discovery that there are infinitely many more diameters of circles than centres of circles implies the discovery that there are infinitely many more real numbers than natural numbers. Probably Bolzano has mentioned this in his collected works, but since they cover more than one meter in the book shelf I am not eager to search for. $\endgroup$ – Otto Jun 26 '17 at 15:18
  • $\begingroup$ I don't know hot to interpret the statement that there are infinitely more diameters than centres in order to deduce the other result from that. If you can expand I would definitely upvote. $\endgroup$ – Michael Bächtold Jun 26 '17 at 15:40

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