Editing Burali-Forti paradox

{{Short description|Paradox in set theory}}
{{No footnotes|date=November 2020}}
In [[set theory]], a field of [[mathematics]], the '''Burali-Forti paradox''' demonstrates that constructing "the set of all [[ordinal number]]s" 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.

==Stated in terms of von Neumann ordinals==

We will prove this by contradiction.

# Let {{mvar|Ω}} be a set consisting of all ordinal numbers.
# {{mvar|Ω}} is [[Transitive set|transitive]] because for every element {{mvar|x}} of {{mvar|Ω}} (which is an ordinal number and can be any ordinal number) and every element {{mvar|y}} of {{mvar|x}} (i.e. under the definition of [[Von Neumann ordinal]]s, for every ordinal number {{math|{{var|y}} < {{var|x}}}}), we have that {{mvar|y}} is an element of {{mvar|Ω}} because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.
# {{mvar|Ω}} is [[Well-order|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 {{mvar|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 {{mvar|Ω}} is an element of {{mvar|Ω}}.
# Under the definition of Von Neumann ordinals, {{math|{{var|Ω}} < {{var|Ω}}}} is the same as {{mvar|Ω}} being an element of {{mvar|Ω}}. This latter statement is proven by step 5.
# But no ordinal class is less than itself, including {{mvar|Ω}} because of step 4 ({{mvar|Ω}} is an ordinal class), i.e. {{math|{{var|Ω}} ≮ {{var|Ω}}}}.

We have deduced two contradictory propositions ({{math|{{var|Ω}} < {{var|Ω}}}} and {{math|{{var|Ω}} ≮ {{var|Ω}}}}) from the sethood of {{mvar|Ω}} and, therefore, disproved that {{mvar|Ω}} 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
[[naive set theory|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 [[axiomatic set theory|axioms for formal set theory]] such as [[Zermelo–Fraenkel set theory|ZF]] and [[Zermelo–Fraenkel set theory|ZFC]] circumvent this antinomy by not allowing the construction of sets using [[unrestricted comprehension|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 axioms{{snd}}specifically Basic Law V{{snd}}in the "Grundgesetze der Arithmetik." Quine's system [[New Foundations]] (NF) uses a [[New Foundations#Burali-Forti paradox|different solution]]. {{harvs|txt|last=Rosser|year=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 (academic)|Hao Wang]].

==See also==
* [[Absolute infinite]]

==References==
{{Reflist}}
{{refbegin}}
* {{citation|first=Cesare|last=Burali-Forti|title= Una questione sui numeri transfiniti|journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=11|pages=154–164|year=1897|doi=10.1007/BF03015911|s2cid=121527917|url=https://zenodo.org/record/2362091}}
* [[Irving Copi]] (1958) "The Burali-Forti Paradox", [[Philosophy of Science (journal)|Philosophy of Science]] 25(4): 281–286, {{doi|10.1086/287617}}
* {{citation|journal=[[Historia Mathematica]]|volume= 8|issue= 3|year= 1981|pages= 319–350|title=Burali-Forti's paradox: A reappraisal of its origins|first1=Gregory H|last1= Moore|first2= Alejandro |last2=Garciadiego
|doi=10.1016/0315-0860(81)90070-7|doi-access= free}}
* {{citation|mr=0006327 |last=Rosser|first= Barkley|title=The Burali-Forti paradox|journal=[[Journal of Symbolic Logic]]|volume= 7|issue=1|year=1942|pages= 1–17|doi=10.2307/2267550|jstor=2267550|s2cid=13389728 }}
{{refend}}

==External links==
*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/paradoxes-contemporary-logic/ Paradoxes and Contemporary Logic]"—by Andrea Cantini.

{{Paradoxes|state=autocollapse}}
{{Set theory}}

[[Category:Ordinal numbers]]
[[Category:Paradoxes of naive set theory]]