## You are here

Homecardinality of the continuum

## Primary tabs

# 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.

Cantorβs 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.) 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.

## Mathematics Subject Classification

03E17*no label found*03E10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

Jun 6

new question: difference of a function and a finite sum by pfb