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.

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

[itex]\Omega < \Omega + 1 \leq \Omega[itex].

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

The Burali-Forti paradox is named after Cesare Burali-Forti, who discovered it in 1897. Burali-Forti was an assistant of Giuseppe Peano in Turin from 1894 to 1896.

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