Countable set

A set is countable if it is finite or can be placed in one-to-one correspondence with a subset of the natural numbers. Infinite countable sets can be listed in a sequence.

Set: A collection of distinct objects, treated as a single mathematical object. Countability classifies sets by whether their elements can be enumerated.
Enumeration: An ordered listing of the elements of a set, usually indexed by natural numbers. A countable set is characterized by the possibility of such a listing.
Integers: The numbers consisting of zero, the positive whole numbers, and their negatives. They are infinite but can be listed by alternating positive and negative values.
Uncountable set: A set that is neither finite nor capable of being listed in a sequence indexed by natural numbers. It marks the boundary beyond countable cardinality.
Natural number: A number used for counting, conventionally including zero or beginning with one. Natural numbers serve as the indices for an enumeration.
Pairing function: A function that assigns a natural number uniquely to each pair of natural numbers. Pairing functions help enumerate objects indexed by two natural numbers.
Rational number: A number expressible as a ratio of two integers with a nonzero denominator. Fractions can be arranged in a sequence despite having infinitely many values.
Cardinality: A measure of the number of elements in a set, including infinite sets. Countability is a basic distinction between possible cardinalities.
Sequence: An ordered list of terms indexed by consecutive integers or natural numbers. An infinite countable set can be listed as a sequence.
Diagonal argument: A proof technique that constructs an object differing from each item in a proposed list at a selected position. It shows that some infinite sets cannot be enumerated.

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