cardinality of the continuum

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

Cantor’s diagonal argument shows that $𝔠$ is uncountable. Furthermore, it can be shown that $ℝ$ is equinumerous with the power set of $\mathbb{N}$, so $𝔠=2^{{\mathrm{\aleph }}_0}$. It can also be shown that $𝔠$ has uncountable cofinality.

It can also be shown that

$$𝔠={𝔠}^{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_0𝔠=𝔠𝔠=𝔠+\kappa ={𝔠}^n$$

for all finite cardinals $n\ge 1$ and all cardinals $\kappa \le 𝔠$. See the article on cardinal arithmetic for some of the basic facts underlying these equalities.

There are many properties of $𝔠$ 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 $𝔠={\mathrm{\aleph }}_n$ 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., $𝔠\ne {\mathrm{\aleph }}_{\omega }$. In particular, $𝔠$ could be either ${\mathrm{\aleph }}_1$ or ${\mathrm{\aleph }}_{{\omega }_1}$, 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