| dbo:abstract
|
- Axiom omezené velikosti (také axiom omezené mohutnosti) je matematické tvrzení z oblasti teorie množin, které je ekvivalentní s axiomem silného výběru. (cs)
- In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V. Von Neumann's axiom implies the axioms of replacement, separation, union, and global choice. It is equivalent to the combination of replacement, union, and global choice in Von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory. Later expositions of class theories—such as those of Paul Bernays, Kurt Gödel, and John L. Kelley—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom. In 1930, Ernst Zermelo defined models of set theory satisfying the axiom of limitation of size. Abraham Fraenkel and Azriel Lévy have stated that the axiom of limitation of size does not capture all of the "limitation of size doctrine" because it does not imply the power set axiom. Michael Hallett has argued that the limitation of size doctrine does not justify the power set axiom and that "von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets." (en)
- En théorie des ensembles, plus précisément en théorie des classes, l'axiome de limitation de taille a été proposé par John von Neumann dans le cadre de sa théorie des classes. Il formalise en partie le principe de limitation de taille (traduction de l'anglais limitation of size), l'un des principes énoncés par Bertrand Russell pour développer la théorie des ensembles en évitant les paradoxes, et qui reprend des idées de Georg Cantor. Le principe est que certaines collections d'objets (d'ensembles en particulier) ont une taille trop grande pour constituer des ensembles, l'univers ensembliste et la classe des ordinaux en particulier. L'axiome de limitation de taille affirme, en substance, qu'une classe, une collection bien définie par une propriété, n'est pas un ensemble si et seulement si elle est équipotente à l'univers ensembliste tout entier, c'est-à-dire qu'elle peut être mise en relation de façon bijective avec celui-ci. Il permet de remplacer l'axiome de séparation, l'axiome de spécialisation et, de façon plus surprenante, l'axiome du choix — il a en fait pour conséquence le principe du choix (ou axiome du choix global), l'axiome du choix sur tout l'univers ensembliste. (fr)
- 在类理论中，大小限制公理声称对于任何类 C，C 是真類(不可以是其他类的元素的类)，当且仅当冯·诺伊曼全集 V (所有集合的类)能一一映射到 C。 这个公理由冯·诺伊曼提出。它蕴涵了分类公理模式、替代公理模式和全局选择公理。大小限制公理蕴涵全局选择公理是因为序数的类不是集合，因此有从全集到序数们的单射。所以集合的全集是良序的。 (zh)
|
| dbo:thumbnail
| |
| dbo:wikiPageExternalLink
| |
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 48060 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:drop
| |
| dbp:proof
|
- To prove the direction, let be a class and be a one-to-one correspondence from to Since maps onto the axiom of limitation of size implies that is a proper class.
To prove the direction, let be a proper class. We will define well-ordered classes and and construct order isomorphisms between and Then the order isomorphism from to is a one-to-one correspondence between and
It was proved above that the axiom of limitation of size implies that there is a function that maps onto Also, was defined as a subclass of that is a one-to-one correspondence between and It defines a well-ordering on if Therefore, is an order isomorphism from to
If is well-ordered class, its proper initial segments are the classes where Now has the property that all of its proper initial segments are sets. Since this property holds for The order isomorphism implies that this property holds for Since this property holds for
To obtain an order isomorphism from to the following theorem is used: If is a proper class and the proper initial segments of are sets, then there is an order isomorphism from to Since and satisfy the theorem's hypothesis, there are order isomorphisms and Therefore, the order isomorphism is a one-to-one correspondence between and (en)
|
| dbp:title
|
- Proof that the axiom of limitation of size implies von Neumann's 1923 axiom (en)
|
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| rdf:type
| |
| rdfs:comment
|
- Axiom omezené velikosti (také axiom omezené mohutnosti) je matematické tvrzení z oblasti teorie množin, které je ekvivalentní s axiomem silného výběru. (cs)
- 在类理论中，大小限制公理声称对于任何类 C，C 是真類(不可以是其他类的元素的类)，当且仅当冯·诺伊曼全集 V (所有集合的类)能一一映射到 C。 这个公理由冯·诺伊曼提出。它蕴涵了分类公理模式、替代公理模式和全局选择公理。大小限制公理蕴涵全局选择公理是因为序数的类不是集合，因此有从全集到序数们的单射。所以集合的全集是良序的。 (zh)
- In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only (en)
- En théorie des ensembles, plus précisément en théorie des classes, l'axiome de limitation de taille a été proposé par John von Neumann dans le cadre de sa théorie des classes. Il formalise en partie le principe de limitation de taille (traduction de l'anglais limitation of size), l'un des principes énoncés par Bertrand Russell pour développer la théorie des ensembles en évitant les paradoxes, et qui reprend des idées de Georg Cantor. Le principe est que certaines collections d'objets (d'ensembles en particulier) ont une taille trop grande pour constituer des ensembles, l'univers ensembliste et la classe des ordinaux en particulier. L'axiome de limitation de taille affirme, en substance, qu'une classe, une collection bien définie par une propriété, n'est pas un ensemble si et seulement si e (fr)
|
| rdfs:label
|
- Axiom omezené velikosti (cs)
- Axiom of limitation of size (en)
- Axiome de limitation de taille (fr)
- 大小限制公理 (zh)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:depiction
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
