Editing Burali-Forti paradox (section)

==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]].