PlanetMath: cardinality of the continuum

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cardinality of the continuum

(Definition)

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

$\displaystyle \mathfrak{c}=\mathfrak{c}^{\aleph_0}=\aleph_0\mathfrak{c}=\mathfrak{c}\mathfrak{c} =\mathfrak{c}+\kappa=\mathfrak{c}^n$

for all finite cardinals $n\ge1$ and all cardinals $\kappa\le\mathfrak{c}$ . 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 , the equality $\mathfrak{c}=\aleph_n$ is independent of ZFC. (The case 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.

"cardinality of the continuum" is owned by yark.

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See Also: cardinal number, cardinal arithmetic

Other names:

cardinal of the continuum, cardinal number of the continuum

Also defines:

continuum many

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Cross-references: singular cardinal, regular cardinal, limit cardinal, successor cardinal, alephs, natural number, consistent, ZFC, equalities, cardinal arithmetic, cardinals, finite, cofinality, power set, uncountable, Cantor's diagonal argument, real numbers, cardinality
There are 7 references to this entry.

This is version 16 of cardinality of the continuum, born on 2004-03-15, modified 2008-11-06.
Object id is 5708, canonical name is CardinalityOfTheContinuum.
Accessed 5427 times total.

Classification:

AMS MSC:	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
	03E17 (Mathematical logic and foundations :: Set theory :: Cardinal characteristics of the continuum)

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