Burali-Forti paradox

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

<math>\Omega < \Omega + 1 \leq \Omega<math>.

Modern axiomatic set theory circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets which have property <math>P<math>", 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 assistent of Giuseppe Peano in Turin from 1894 to 1896.