- Positive set theory
In
mathematical logic , positive set theory is the name for a class of alternative set theories in which theaxiom of comprehension *"x mid phi} exists"
holds for at least the positive formulas phi (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 GPK^+_{infty} of Olivier Esser consists of the following axioms:
* The
axiom of extensionality : x=y Leftrightarrowforall a, (ain x Leftrightarrow ain y).
* Theaxiom of empty set : there exists a set emptyset such that eg exists_x xinemptyset (this axiom can be neatly dispensed with if a false formula perp is included as a positive formula).
* The axiom of generalized positive comprehension: if phi is a formula in predicate logic using only vee, wedge, exists, forall, , and in, then the set of all x such that phi(x) is also a set. Quantification (forall, exists) may be bounded.
** Note that negation is specifically not permitted.
* The axiom of closure: for every formula phi(x), a set exists which is the intersection of all sets which contain every "x" such that phi(x); this is called the closure of x mid phi(x)} 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.
* Theaxiom of infinity : thevon Neumann ordinal omega exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of omega exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that omega contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength ofMorse–Kelley set theory with the proper class ordinal aweakly compact cardinal .Interesting properties
* The
universal set is a proper set in this theory.
* The sets of this theory are the collections of sets which are closed under a certaintopology on the classes.
* The theory can interpretZFC (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory (Morse-Kelley set theory with the proper class ordinal aweakly compact cardinal ).Researchers
*
Isaac Malitz originally introduced Positive Set Theory in his 1976 PhD Thesis at UCLA
*Alonzo Church was the chairman of the committee supervising the aforementioned thesis
*Olivier Esser seems to be the most active in this field.See also
*
New Foundations by QuineReferences
*citation|id=MR|1669902
last=Esser|first= Olivier
title=On the consistency of a positive theory.
journal=MLQ Math. Log. Q.|volume= 45 |year=1999|issue= 1|pages= 105-116
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