Georg Cantor

"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."

"If we have only to classify a finite number of objects, it is easy to preserve these classifications without change. If the number of objects is indefinite, ...[i.e.,] if we are constantly liable to find new and unforeseen objects springing up, it may happen that the appearance of a new object will oblige us to modify the classification, and it is thus that we are exposed to antinomies. There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. It is true that Cantorism has been useful, but that was when it was applied to a real problem, whose terms were clearly defined, and then it was possible to advance without danger. Like the Cantorians, the logicians have forgotten the fact, and they have met with the same difficulties. ...[B]elief in an actual infinity is essential in the Russellian logic, and this... distinguishes it from the Hilbertian logic. Hilbert takes the... view of extension... to avoid the Cantorian antimonies. Russell takes the... view of comprehension... to regard the infinite as actual. And we have not only infinite classes; when we pass from the genus to the species... the number of conditions is still infinite, for they generally express that the object... is in... relation with all the objects of an infinite class. But all this is ."

"[T]here exist no other sets than finite and denumerably infinite sets and continua... [I]n mathematics we can create only finite sequences, further by means of... and so on the order type ω, but only consisting of equal elements... but no other sets. Cantor and his disciples... think they have knowledge of all sorts of further sets; their fundamental principle... comes to about the same as the axiomaticians. ...[T]his principle is unjustified and... we assert that the several paradoxes of the Mengenlehre... have no right to exist... [I]t would have been the duty of Cantorians, immediately to reject a notion which gives rise to contradictions, because it is... not built... mathematically."

"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."

"I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God. I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures."

"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."

"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."

"As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche]."

"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."

"What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers."

"Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real."

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