2aleph0

2^{aleph_0} (often denoted c) is a central cardinal number in set theory. It is defined as the cardinality of the power set of the natural numbers, P(N). Equivalently, it is the cardinality of the real numbers, since there is a bijection between P(N) and R through binary expansions or other standard encodings. Thus 2^{aleph_0} is also described as the cardinality of the continuum.

As a cardinal, 2^{aleph_0} is strictly larger than aleph_0, the cardinality of the natural numbers, by Cantor’s theorem. It is the first uncountable cardinal, though whether it equals aleph_1 depends on the truth of the continuum hypothesis (CH).

The continuum hypothesis posits that 2^{aleph_0} = aleph_1. CH is famously independent of Zermelo-Fraenkel set theory with the axiom of choice (ZFC): Gödel showed CH cannot be disproven from ZFC, while Cohen showed CH cannot be proven from ZFC. Consequently, in some models of ZFC CH holds, while in others 2^{aleph_0} can be larger than aleph_1 (consistently up to any regular cardinal or beyond, subject to certain constraints from set theory).

Beyond CH, 2^{aleph_0} plays a key role in cardinal arithmetic and the study of the continuum, influencing analysis, topology, and descriptive set theory. It is often referred to simply as the continuum, highlighting its foundational status as the size of the real-number line.