In re mathematica ars proponendi pluris facienda est quam solvendi.

Georg Cantor— In mathematics the art of asking questions is more valuable than solving problems.Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.

Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!

Georg Cantor— Letter (1885), written after Gösta Mittag-Leffler persuaded him to withdraw a submission to Mittag-Leffler's journal

The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false.

Georg Cantor— "Über unendliche, lineare Punktmannigfaltigkeiten" in

Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde.

Georg Cantor— However, he was born in Copenhagen, of Jewish parents, of the Portuguese Jewish community there.

Of his father. In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, 1934, p. 306)

Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.

Georg Cantor— Letter to Richard Dedekind (1899), as translated in

A set is a Many that allows itself to be thought of as a One.

Georg Cantor— As quoted in

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo , where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

Georg Cantor— As quoted in

That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices.

Georg Cantor— Letter to Gustac Enestrom, as quoted in

What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite ), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers .

Georg Cantor— As quoted in

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

Georg Cantor— As quoted in

I entertain no doubts as to the truths of the transfinites, which I recognized with God "s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.

Georg Cantor— As quoted in

The essence of mathematics lies entirely in its freedom.

Georg Cantor— Variant translation: The essence of mathematics is in its freedom.

I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.

Georg Cantor—

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.

Georg Cantor— As quoted in