2alephnull

2^{aleph_0}, commonly denoted c and referred to as the continuum, is a central cardinal in set theory. It is the cardinality of the set of real numbers, and is also the cardinality of the power set of the natural numbers, P(N), as well as the set of all infinite binary sequences {0,1}^N. In all these senses, 2^{aleph_0} measures the size of continuous mathematical objects.

Cantor’s diagonal argument shows that 2^{aleph_0} is strictly larger than aleph_0, meaning the real numbers are uncountable. The continuum is the first cardinal above the countable ones in the standard hierarchy, though its exact position depends on the underlying set theory.

The continuum hypothesis (CH) is the assertion that 2^{aleph_0} = aleph_1, i.e., that the continuum is the first uncountable cardinal. CH is independent of ZFC: Gödel showed CH is consistent with ZFC if ZFC itself is consistent, while Cohen showed that the negation of CH is also consistent with ZFC. Consequently, ZFC does not decide the exact value of 2^{aleph_0}.

In practice, 2^{aleph_0} is often treated as the cardinality of the real line, the continuum, and serves as a baseline size for many constructions in analysis, topology, and logic. Depending on the chosen axioms beyond ZFC, the continuum can equal aleph_1 or be strictly larger, with its precise value varying across models of set theory.