Burali-Forti paradox

• Burali-Forti's paradox

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.

Burali-Forti paradox

1. (set theory) The paradox that supposing the existence of a set of all ordinal numbers leads to a contradiction; construed as meaning that it is not a properly defined set.
   • 1984, Michael Hallett, Cantorian Set Theory and Limitation of Size, Oxford University Press (Clarendon Press), 1986, Paperback, page 186,

     Like them, Mirimanoff concentrates on the Burali-Forti paradox, and like Russell's analysis before, Mirimanoff shows how. in terms of size, the Burali-Forti paradox is basic and that if we solve this the other paradoxes will be solved too.

   • 1994 [Routledge], Ivor Grattan-Guinness (editor), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Volume 1, 2003, Johns Hopkins University Press, Paperback, page 632,

     In the first place, Berry rejected Russell's solution to the Burali-Forti paradox, claiming that it was easy to prove that the set of all ordinal numbers was a well-ordered set (and that Cantor had actually done it).

   • 2002, Marcus Giaquinto, The Search for Certainty: A Philosophical Account of Foundations of Mathematics, Oxford University Press (Clarendon Press), page 37,

     The Burali-Forti paradox was discovered by Cantor in 1895 and Burali-Forti in 1897, but was not regarded by them as a paradox.

In modern axiomatic set theory, the paradox is resolved by disallowing certain ways of defining sets (specifically, the use of unrestricted comprehension terms such as "all sets with property P").

The paradox may be enunciated as follows:

• Associate with each well-ordering an object called its order type (the order types are the ordinal numbers).
• The order types (ordinal numbers) are themselves well-ordered in a natural way, so this well-ordering must itself have an order type, ${\displaystyle \Omega }$ .
• As can be shown (in naive set theory, as well as the now standard ZFC), the order type of the ordinal numbers less than a fixed ${\displaystyle \alpha }$ is ${\displaystyle \alpha }$ .
• Therefore the order type of all ordinal numbers less than ${\displaystyle \Omega }$ is ${\displaystyle \Omega }$ .
• Consequently, ${\displaystyle \Omega }$ , being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals; but the latter is ${\displaystyle \Omega }$ itself, by definition.
• This is a contradiction.

• Russell's paradox