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'cardinality of the continuum'
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| Title of object: |
cardinality of the continuum |
| Canonical Name: |
CardinalityOfTheContinuum |
| Type: |
Definition |
| Created on: |
2004-03-15 07:35:23 |
| Modified on: |
2006-12-13 09:42:47 |
| Classification: |
msc:03E10, msc:03E17 |
| Defines: |
continuum many |
| Synonyms: |
cardinality of the continuum=cardinal of the continuum |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\N{\mathbb{N}}
\def\R{\mathbb{R}}
\def\continuum{\mathfrak{c}} |
Content:
\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.
Many properties of $\continuum$ are 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. |
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