Cardinality of the continuum

The cardinality of the real numbers, equal to the cardinality of the power set of the natural numbers and denoted 𝔠 or 2^ℵ₀.

Cardinal number: A mathematical measure of the size of a set, defined by whether it can be paired bijectively with another set. The continuum is a particular cardinal number, compared with others through bijections.
Cantor's theorem: The theorem that every set has strictly smaller cardinality than its power set. Applied to the natural numbers, it shows the continuum exceeds ℵ₀.
Continuum hypothesis: The hypothesis that no cardinality lies strictly between the natural numbers and the real numbers. It asks whether 𝔠 is exactly the next cardinal after ℵ₀.
Space of real sequences: The set of all infinite sequences whose terms are real numbers. This space has cardinality 𝔠, since a sequence of reals can be encoded by a real.
Countable set: A set that is finite or can be placed in one-to-one correspondence with the natural numbers. The continuum exceeds countable cardinality, despite the rationals being dense in the reals.
Binary sequence: A sequence whose entries are drawn from the two symbols 0 and 1. Infinite binary sequences correspond to subsets of the natural numbers and thus have continuum cardinality.
Independence of the continuum hypothesis: The result that the continuum hypothesis is unprovable and irrefutable from the standard ZFC axioms, assuming they are consistent. Its independence means ZFC alone cannot settle the relation between ℵ₀ and 𝔠.
Set of real-valued functions: The collection of all functions from a specified domain into the real numbers. For a countably infinite domain, this function space has cardinality 𝔠.
Bijection: A function that pairs every element of one set with exactly one element of another, and vice versa. Bijections give the precise criterion for the continuum's equality with other cardinalities.
Cantor set: A subset of the unit interval formed by repeatedly removing open middle thirds. Despite containing no interval, this canonical set has the same cardinality as the continuum.

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Martin's axiomRelated: Martin's axiom controls the number of dense sets relative to this cardinal.