In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.
It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor.
Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.
Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its order type in an unspecified way (the order types are the ordinal numbers).
It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed
But the collection of order types in New Foundations (defined as equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of the ordinals less than
", as is possible in naive set theory and as is possible with Gottlob Frege's axioms – specifically Basic Law V – in the "Grundgesetze der Arithmetik."
Rosser (1942) showed that in the original version of Quine's system "Mathematical Logic" (ML), an extension of New Foundations, it is possible to derive the Burali-Forti paradox, showing that this system was contradictory.
Quine's revision of ML following Rosser's discovery does not suffer from this defect, and indeed was subsequently proved equiconsistent with NF by Hao Wang.