Uncountable set

An infinite set that cannot be put in bijection with the natural numbers. Its elements cannot be listed as a sequence indexed by the natural numbers.

Cantor's diagonal argument: A proof technique that constructs an object differing from every entry in a proposed infinite list. Cantor used it to prove that real numbers cannot be completely listed.
Real number: A number on the continuous number line, including rational and irrational numbers. The real numbers are the standard example of an uncountable set.
Natural number: An element of the number system used for counting, conventionally beginning with zero or one. Uncountability is defined by failure to match this set in a bijection.
Continuum hypothesis: The claim that no cardinality lies strictly between that of the natural numbers and the real numbers. It asks whether the real numbers are the smallest uncountable set in cardinality.
Cantor's theorem: The power set of any set has strictly greater cardinality than the set itself. It guarantees uncountable power sets, including the real numbers' power set relationship.
Irrational number: A real number that cannot be expressed as a ratio of two integers. The irrational numbers form an uncountable subset of the real numbers.
Infinite set: A set that is not finite; equivalently, one that can be put in bijection with a proper subset of itself. Uncountable sets are a particular kind of infinite set.
Uncountable union: A union indexed by an uncountable family of sets. Uncountable indexing creates constructions whose size may exceed countable bounds.
Cardinality: A measure of the size of a set defined by bijections with other sets. Uncountability is a cardinality distinction from the natural numbers.
Power set: The set of all subsets of a given set. The power set of the natural numbers is uncountable by Cantor's theorem.

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