“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 From Frege to Gödel : A Source Book in Mathematical Logic, 1879-1931 (1967) by Jean Van Heijenoort, p. 117

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Georg Cantor 27

mathematician, inventor of set theory 1845–1918

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“The transfinite numbers are in a certain sense themselves new irrationalities”

Georg Cantor (1845–1918) mathematician, inventor of set theory

As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Context: 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 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 (1845–1918) mathematician, inventor of set theory

As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178

“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.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)

“In a free Government, the security for civil rights must be the same as that for religious rights. It consists in the one case in the multiplicity of interests, and in the other in the multiplicity of sects. The degree of security in both cases, will depend on the number of interests and sects; and this may be presumed to depend on the extent of country and number of People comprehended under the same Government.”

James Madison (1751–1836) 4th president of the United States (1809 to 1817)

Federalist No. 51 (6 February 1788)
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“The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

Letter to David Hilbert (2 October 1897)
Context: The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. This is the punctum saliens, and I venture to say that this completely certain theorem, provable rigorously from the definition of the totality of all alephs, is the most important and noblest theorem of set theory. One must only understand the expression "finished" correctly. I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements.