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'limit cardinal'
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| Title of object: |
limit cardinal |
| Canonical Name: |
LimitCardinal |
| Type: |
Definition |
| Created on: |
2003-12-01 06:32:53 |
| Modified on: |
2003-12-01 12:56:52 |
| Classification: |
msc:03E10 |
| Defines: |
strong limit cardinal |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
A {\em limit cardinal} is a cardinal $\kappa$ such that $\lambda^+<\kappa$ for every cardinal $\lambda<\kappa$. Here $\lambda^+$ denotes the cardinal successor of $\lambda$. If $2^\lambda<\kappa$ for every cardinal $\lambda<\kappa$, then $\kappa$ is called a {\em strong limit cardinal}. Every strong limit cardinal is a limit cardinal, because $\lambda^+\leq2^\lambda$ holds for every cardinal $\lambda$.
Under GCH every limit cardinal is a strong limit cardinal, because in this case $\lambda^+=2^\lambda$ for every infinite cardinal $\lambda$.
Note that some authors do not count $0$ as a limit cardinal. Some even exclude $\aleph_0$. |
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