Infinity: Countable Set and Continuum Hypothesis

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  • Topic: Countable set, Georg Cantor, Infinity
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  • Published : December 2, 2012
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Infinity has a relationship with mathematics proven by the existence of countable infinity and continuum infinity. Countably infinite numbers is any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set  is given, any other set which can be put into a one-to-one correspondence with  is also countable. Countably infinite sets have cardinal number aleph-0. While on the other hand, continuously infinite numbers, also known as continuum hypothesis, proved by the German mathematician Georg Cantor, states that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers  (the "continuum").

Courant, R. and Robbins, H. "The Denumerability of the Rational Number and the Non-Denumerability of the Continuum." §2.4.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 79-83, 1996. Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 10, 1988.

Cohen, P. J. "The Independence of the Continuum Hypothesis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963. Cohen, P. J. "The Independence of the Continuum Hypothesis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964. Cohen, P. J. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.

Infinity also has a relationship with physics which can be seen and proved by the theoretical applications of physical infinity. The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations. One of the needs of any physical and scientific theory is to give usable formulas that...
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