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 .

Examples of countable sets include the integers , algebraic numbers , and rational numbers . Georg Cantor
showed that the number of real numbers is rigorously
larger than a countably infinite set, and the postulate that this number, the so-called
"continuum ," is equal to aleph-1
is called the continuum hypothesis . Examples
of nondenumerable sets include the real , complex ,
irrational , and transcendental
numbers .

See also Aleph-0 ,

Aleph-1 ,

Cantor Diagonal Method ,

Cardinal
Number ,

Continuum ,

Continuum
Hypothesis ,

Countable Set ,

Hilbert
Hotel ,

Infinite ,

Infinity ,

Uncountably Infinite
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References 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. Referenced on Wolfram|Alpha Countably Infinite
Cite this as:
Weisstein, Eric W. "Countably Infinite."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/CountablyInfinite.html

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