Английская Википедия:Burali-Forti paradox
Шаблон:Short description Шаблон:No footnotes 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.
Stated in terms of von Neumann ordinals
We will prove this by contradiction.
- Let Шаблон:Mvar be a set consisting of all ordinal numbers.
- Шаблон:Mvar is transitive because for every element Шаблон:Mvar of Шаблон:Mvar (which is an ordinal number and can be any ordinal number) and every element Шаблон:Mvar of Шаблон:Mvar (i.e. under the definition of Von Neumann ordinals, for every ordinal number Шаблон:Math), we have that Шаблон:Mvar is an element of Шаблон:Mvar because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.
- Шаблон:Mvar 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 Шаблон:Mvar 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 Шаблон:Mvar is an element of Шаблон:Mvar.
- Under the definition of Von Neumann ordinals, Шаблон:Math is the same as Шаблон:Mvar being an element of Шаблон:Mvar. This latter statement is proven by step 5.
- But no ordinal class is less than itself, including Шаблон:Mvar because of step 4 (Шаблон:Mvar is an ordinal class), i.e. Шаблон:Math.
We have deduced two contradictory propositions (Шаблон:Math and Шаблон:Math) from the sethood of Шаблон:Mvar and, therefore, disproved that Шаблон:Mvar is a set.
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 <math>\Omega</math>. 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 <math>\alpha</math> is <math>\alpha</math> itself. So the order type of all ordinal numbers less than <math>\Omega</math> is <math>\Omega</math> itself. But this means that <math>\Omega</math>, 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 <math>\Omega</math> itself by definition. This is a contradiction.
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 <math>\alpha</math> is <math>\alpha</math> itself must be true. The collection of von Neumann ordinals, like the collection in the Russell paradox, cannot be a set in any set theory with classical logic. 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 <math>\Omega</math> turns out not to be <math>\Omega</math>.
Resolutions of the paradox
Modern axioms for formal set theory such as ZF and ZFC circumvent this antinomy by not allowing the construction of sets using terms like "all sets with the property <math>P</math>", as is possible in naive set theory and as is possible with Gottlob Frege's axiomsШаблон:Sndspecifically Basic Law VШаблон:Sndin the "Grundgesetze der Arithmetik." Quine's system New Foundations (NF) uses a different solution. Шаблон:Harvs 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.
See also
References
- Шаблон:Citation
- Irving Copi (1958) "The Burali-Forti Paradox", Philosophy of Science 25(4): 281–286, Шаблон:Doi
- Шаблон:Citation
- Шаблон:Citation
External links
- Stanford Encyclopedia of Philosophy: "Paradoxes and Contemporary Logic"—by Andrea Cantini.
Шаблон:Paradoxes Шаблон:Set theory
