# cardinality of the continuum

The cardinality of the continuum, often denoted by $\mathfrak{c}$, is the cardinality of the set $\mathbb{R}$ of real numbers. A set of cardinality $\mathfrak{c}$ is said to have continuum many elements.

Cantor’s diagonal argument shows that $\mathfrak{c}$ is uncountable. Furthermore, it can be shown that $\mathbb{R}$ is equinumerous with the power set of $\mathbb{N}$, so $\mathfrak{c}=2^{\aleph_{0}}$. It can also be shown that $\mathfrak{c}$ has uncountable cofinality.

It can also be shown that

 $\mathfrak{c}=\mathfrak{c}^{\aleph_{0}}=\aleph_{0}\mathfrak{c}=\mathfrak{c}% \mathfrak{c}=\mathfrak{c}+\kappa=\mathfrak{c}^{n}$

for all finite cardinals $n\geq 1$ and all cardinals $\kappa\leq\mathfrak{c}$. See the article on cardinal arithmetic for some of the basic facts underlying these equalities.

There are many properties of $\mathfrak{c}$ 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 $\mathfrak{c}=\aleph_{n}$ is independent of ZFC. (The case $n=1$ is the well-known Continuum Hypothesis (http://planetmath.org/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., $\mathfrak{c}\neq\aleph_{\omega}$. In particular, $\mathfrak{c}$ 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.

 Title cardinality of the continuum Canonical name CardinalityOfTheContinuum Date of creation 2013-03-22 14:15:33 Last modified on 2013-03-22 14:15:33 Owner yark (2760) Last modified by yark (2760) Numerical id 19 Author yark (2760) Entry type Definition Classification msc 03E17 Classification msc 03E10 Synonym cardinal of the continuum Synonym cardinal number of the continuum Related topic CardinalNumber Related topic CardinalArithmetic Defines continuum many