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- Axiom_of_limitation_of_size abstract "In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class which is not a set (an element of other classes), if and only if it can be mapped onto the class V of all sets.This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union at one stroke. The axiom of limitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is a surjection from the ordinals to the universe, thus an injection from the universe to the ordinals, that is, the universe of sets is well-ordered.Together the axiom of replacement and the axiom of global choice (with the other axioms of von Neumann–Bernays–Gödel set theory) imply this axiom. This axiom can then replace replacement, global choice, specification and union in von Neumann–Bernays–Gödel or Morse–Kelley set theory.However, the axiom of replacement and the usual axiom of choice (with the other axioms of von Neumann–Bernays–Gödel set theory) do not imply von Neumann's axiom. In 1964, Easton used forcing to build a model that satisfies the axioms of von Neumann–Bernays–Gödel set theory with one exception: the axiom of global choice is replaced by the axiom of choice. In Easton's model, the axiom of limitation of size fails dramatically: the universe of sets cannot even be linearly ordered.It can be shown that a class is a proper class if and only if it is equinumerous to V, but von Neumann's axiom does not capture all of the "limitation of size doctrine", because the axiom of power set is not a consequence of it. Later expositions of class theories (Bernays, Gödel, Kelley, ...) generally use replacement and a form of the axiom of choice rather than the axiom of limitation of size.".
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- Axiom_of_limitation_of_size wikiPageExternalLink 9.pdf.
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- Axiom_of_limitation_of_size subject Category:Axioms_of_set_theory.
- Axiom_of_limitation_of_size subject Category:Wellfoundedness.
- Axiom_of_limitation_of_size type Abstraction100002137.
- Axiom_of_limitation_of_size type AuditoryCommunication107109019.
- Axiom_of_limitation_of_size type AxiomsOfSetTheory.
- Axiom_of_limitation_of_size type Communication100033020.
- Axiom_of_limitation_of_size type Maxim107152948.
- Axiom_of_limitation_of_size type Saying107151380.
- Axiom_of_limitation_of_size type Speech107109196.
- Axiom_of_limitation_of_size comment "In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class which is not a set (an element of other classes), if and only if it can be mapped onto the class V of all sets.This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union at one stroke.".
- Axiom_of_limitation_of_size label "Axiom of limitation of size".
- Axiom_of_limitation_of_size label "Axiome de limitation de taille".
- Axiom_of_limitation_of_size label "大小限制公理".
- Axiom_of_limitation_of_size sameAs Axiom_omezené_velikosti.
- Axiom_of_limitation_of_size sameAs Axiome_de_limitation_de_taille.
- Axiom_of_limitation_of_size sameAs m.0dfj6g.
- Axiom_of_limitation_of_size sameAs Q836475.
- Axiom_of_limitation_of_size sameAs Q836475.
- Axiom_of_limitation_of_size sameAs Axiom_of_limitation_of_size.
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- Axiom_of_limitation_of_size isPrimaryTopicOf Axiom_of_limitation_of_size.