“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.”
As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178
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Georg Cantor 27
mathematician, inventor of set theory 1845–1918Related quotes
“The world's nature is a harmonious compound of infinite and finite elements”
The Life of Pythagoras (1919)
Context: Fragment 1. (Stob.21.7; Diog.#.8.85) The world's nature is a harmonious compound of infinite and finite elements; similar is the totality of the world in itself, and of all it contains.
b. All beings are necessarily finite or infinite, or simultaneously finite and infinite; but they could not all be infinite only.
Source: The Limits of Evolution, and Other Essays, Illustrating the Metaphysical Theory of Personal Ideaalism (1905), Appendix E: Reply to Criticisms of Mr. J.M.E. McTaggert, p.421-2
Mathematical Problems (1900)
As quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
Context: 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.
“It is the infinite alone that cannot be attained, for if it could it would become finite.”
Thoughts on Art and Life, by Leonardo da Vinci http://www.gutenberg.org/ebooks/29904, (1906)
