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'Burali-Forti paradox'
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| Title of object: |
Burali-Forti paradox |
| Canonical Name: |
BuraliFortiParadox |
| Type: |
Definition |
| Created on: |
2002-09-28 19:13:57 |
| Modified on: |
2006-08-16 18:02:34 |
| Classification: |
msc:03-00 |
Revision comment (for changes between this and next version):
| Changes for correction #8912 ('linking (even)'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
%\PMlinkescapeword{theory} |
Content:
The \emph{Burali-Forti} paradox demonstrates that the class of all ordinals is not a set. If there were a set of all ordinals, $Ord$, then it would follow that $Ord$ was itself an ordinal, and therefore that $Ord\in Ord$. Even if sets in general are allowed to contain themselves, ordinals cannot since they are defined so that $\in$ is well founded over them.
This paradox is similar to both Russell's paradox and Cantor's paradox, although it predates both. All of these paradoxes prove that a certain object is ``too large'' to be a set. |
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