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Burali-Forti paradox

The Burali-Forti paradox demonstrates that the ordinal numbers, unlike the natural numbers, do not form a set. The ordinal numbers can be defined as the set of all their predecessors. Thus,

0 is defined as {}, the empty set
1 is defined as {0} which can be written as {{}}
2 is defined as {0, 1} which can be written as {{}, {{}}}
3 is defined as {0, 1, 2} which can be written as {{}, {{}}, {{}, {{}}}}
in general, n is defined as {0, 1, 2, ... n-1}

By this definition, if the ordinal numbers formed a set, that set would then be an ordinal number greater than any number in the set. This contradicts the assertion that the set contains all ordinal numbers.

Although the ordinal numbers do not form a set, they can be regarded as a collection called a class.


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