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 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.
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Stated in terms of von Neumann ordinals
We will prove this by reductio ad absurdum.
- Let
Ω be a set that contains all ordinal numbers. -
Ω is transitive because for every elementx of it (which is an ordinal number and can be any ordinal number) every elementy ofx (i.e. under the definition of Von Neumann ordinals, for every ordinal numbery < x ) we have thaty is an element ofΩ because any ordinal number only contains ordinal numbers, by the definition of this ordinal construction. -
Ω is well ordered by the membership relation because all its elements are also well ordered by this relation. - So, by steps 2 and 3, we have that
Ω is an ordinal class and also, by step 1, an ordinal number, because all ordinal classes that are sets are also ordinal numbers. - This implies that
Ω is an element ofΩ . - Under the definition of Von Neumann ordinals,
Ω < Ω is the same asΩ being an element ofΩ . This latter statement is proven by step 5. - But we have that no ordinal class is less than itself, including
Ω because of step 4 (Ω is an ordinal class), i.e.¬ Ω < Ω .
We've deduced two contradictory propositions (
Stated more generally
The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. 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). The order types (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have an order type
If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed
Resolution of the paradox
Modern axiomatic set theory such as ZF and ZFC circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets with the property