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