A set is countable if it is either finite or can be put in one-to-one correspondence with the natural numbers, equivalently if there exists an injective function from the set into the natural numbers. A countable set that is not finite is called countably infinite, such as the set of all natural numbers or all rational numbers, while the set of real numbers is an example of an uncountable set. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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Cardinality, Concepts Countability is exactly a classification by cardinality: a countable set is one whose cardinality is finite or matches the cardinality of the natural numbers.
Additional Source Countable Set (Wikipedia)Lead section
Infinity, Concepts A countably infinite set is the smallest kind of infinite set, put in correspondence with the natural numbers rather than being merely finite, so the concept names one specific way a set can be infinite.
Additional Source Countable Set (Wikipedia)Lead section
Sets, Concepts Countability is a classification of sets by whether they can be put in one to one correspondence with the natural numbers, so the concept is a property of the set concept.
Additional Source Countable Set (Wikipedia)Lead section
Sources
1. Countable Set (Wikipedia)
Wikimedia FoundationLead section
A mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers
History section, first paragraph
In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable.
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