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- Positive_set_theory abstract "In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension"exists"holds for at least the positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification).Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact.The set theory of Olivier Esser consists of the following axioms: The axiom of extensionality: . The axiom of empty set: there exists a set such that (this axiom can be neatly dispensed with if a false formula is included as a positive formula). The axiom of generalized positive comprehension: if is a formula in predicate logic using only , , , , , and , then the set of all such that is also a set. Quantification (, ) may be bounded. Note that negation is specifically not permitted. The axiom of closure: for every formula , a set exists which is the intersection of all sets which contain every x such that this is called the closure of and is written in any of the various ways that topological closures can be presented. This can be put more briefly if class language is allowed (any condition on sets defining a class as in NBG): for any class C there is a set which is the intersection of all sets which contain C as a subclass. This is obviously a reasonable principle if the sets are understood as closed classes in a topology. The axiom of infinity: the von Neumann ordinal exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal.".
- Positive_set_theory wikiPageID "1023857".
- Positive_set_theory wikiPageRevisionID "546492731".
- Positive_set_theory hasPhotoCollection Positive_set_theory.
- Positive_set_theory subject Category:Systems_of_set_theory.
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- Positive_set_theory type Instrumentality103575240.
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- Positive_set_theory type System104377057.
- Positive_set_theory type SystemsOfSetTheory.
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- Positive_set_theory comment "In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension"exists"holds for at least the positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification).Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology.".
- Positive_set_theory label "Positive set theory".
- Positive_set_theory label "正集合论".
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- Positive_set_theory sameAs Q7233294.
- Positive_set_theory sameAs Q7233294.
- Positive_set_theory sameAs Positive_set_theory.
- Positive_set_theory wasDerivedFrom Positive_set_theory?oldid=546492731.
- Positive_set_theory isPrimaryTopicOf Positive_set_theory.