cardinality of the continuum CardinalityOfTheContinuum 2004-03-15 07:35:23 2008-11-06 05:20:07 Definition cardinal number continuum many \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\N{\mathbb{N}} \def\R{\mathbb{R}} \def\continuum{\mathfrak{c}} \PMlinkescapeword{independent} \PMlinkescapeword{nor} \PMlinkescapeword{properties} The \emph{cardinality of the continuum}, often denoted by $\continuum$, is the cardinality of the set $\R$ of real numbers. A set of cardinality $\continuum$ is said to have \emph{continuum many} elements. Cantor's diagonal argument shows that $\continuum$ is uncountable. Furthermore, it can be shown that $\R$ is equinumerous with the power set of $\N$, so $\continuum=2^{\aleph_0}$. It can also be shown that $\continuum$ has uncountable cofinality. It can also be shown that $$\continuum=\continuum^{\aleph_0}=\aleph_0\continuum=\continuum\continuum =\continuum+\kappa=\continuum^n$$ for all finite cardinals $n\ge1$ and all cardinals $\kappa\le\continuum$. See the article on cardinal arithmetic for some of the basic facts underlying these equalities. There are many properties of $\continuum$ that independent of ZFC, that is, they can neither be proved nor disproved in ZFC, assuming that ZF is consistent. For example, for every nonzero natural number $n$, the equality $\continuum=\aleph_n$ is independent of ZFC. (The case $n=1$ is the well-known \PMlinkname{Continuum Hypothesis}{ContinuumHypothesis}.) The same is true for most other alephs, although in some cases equality can be ruled out on the grounds of cofinality, e.g., $\continuum\neq\aleph_\omega$. In particular, $\continuum$ could be either $\aleph_1$ or $\aleph_{\omega_1}$, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.