Burali-Forti paradox: Difference between revisions

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.

We will prove this by reductio ad absurdumcontradiction.

# Let <math>\Omega</math>{{mvar|Ω}} be a set thatconsisting containsof all ordinal numbers.

# <math>\Omega</math>{{mvar|Ω}} is [[Transitive set|transitive]] because for every element <math>{{mvar|x</math>}} of <math>\Omega</math>{{mvar|Ω}} (which is an ordinal number and can be any ordinal number) and every element <math>{{mvar|y</math>}} of <math>{{mvar|x</math>}} (i.e. under the definition of [[Von Neumann ordinal]]s, for every ordinal number <{{math>|{{var|y}} < {{var|x</math>}}}}), we have that <math>{{mvar|y</math>}} is an element of <math>\Omega</math>{{mvar|Ω}} because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.

# <math>\Omega</math>{{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 <math>\Omega</math>{{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 <math>\Omega</math>{{mvar|Ω}} is an element of <math>\Omega</math>{{mvar|Ω}}.

# Under the definition of Von Neumann ordinals, <{{math>\Omega|{{var|Ω}} < \Omega</math>{{var|Ω}}}} is the same as <math>\Omega</math>{{mvar|Ω}} being an element of <math>\Omega</math>{{mvar|Ω}}. This latter statement is proven by step 5.

# But we have that no ordinal class is less than itself, including <math>\Omega</math>{{mvar|Ω}} because of step 4 (<math>\Omega</math>{{mvar|Ω}} is an ordinal class), i.e. <{{math>\lnot|{{var|Ω}} (\Omega≮ < \Omega)</math>{{var|Ω}}}}.

We've have deduced two contradictory propositions (<{{math>\Omega|{{var|Ω}} < \Omega</math>{{var|Ω}}}} and <{{math>\lnot|{{var|Ω}} (\Omega <≮ \Omega)</math>{{var|Ω}}}}) from the sethood of <math>\Omega</math>{{mvar|Ω}} and, therefore, disproved that <math>\Omega</math>{{mvar|Ω}} is a set.

Modern [[axiomatic set theory|axioms for formal set theory]] such as ZF and 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#How NF.28U.29 avoids the setBurali-theoreticForti paradoxesparadox|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]].

* [[Absolute Infiniteinfinite]]

* {{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/files/article.pdf}}

* {{citation|journal=[[Historia Mathematica]]|volume= 8|issue= 3|year= 1981|pages= 319–350|title=Burali-Forti's paradox: A reappraisal of its origins|firstfirst1=Gregory H|lastlast1= Moore|first2= Alejandro |last2=Garciadiego

* {{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 }}