Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naïvely constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.

Related Topics:
Set theory - Mathematics - Ordinal number - Antinomy

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The reason is that the set of all ordinal numbers Omega carries all properties of an ordinal number and would have to be considered an ordinal number itself. Then, we can construct its successor Omega + 1, which is strictly greater than Omega. However, this ordinal number must be an element of Omega since Omega contains all ordinal numbers, and we arrive at

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:Omega < Omega + 1 leq Omega.

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Modern axiomatic set theory circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets which have property P", as it was for example possible in Gottlob Frege's axiom system.

Related Topics:
Axiomatic set theory - Comprehension terms - Gottlob Frege

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The Burali-Forti paradox is named after Cesare Burali-Forti, who discovered it in 1897.

Related Topics:
Cesare Burali-Forti - 1897

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