Burali–Forti paradox: Difference between revisions
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The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes). |
The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes). |
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Latest revision as of 16:55, 25 March 2024
The Burali–Forti paradox refers to the theorem that there is no set containing all von Neumann ordinals. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes).