Positive set theory
The set theory formula_2 of Olivier Esser consists of the following axioms: formula_3 formula_4
where formula_1 is a "positive formula". A positive formula uses only the logical constants formula_6 but not formula_7. formula_8
where formula_1 is a formula. That is, for every formula formula_1, the intersection of all sets which contain every formula_11 such that formula_12 exists. This is called the closure of formula_13 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 a reasonable principle if the sets are understood as closed classes in a topology. The von Neumann ordinal formula_14 exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of formula_14 exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that formula_14 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.
Axioms
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas formula_1 (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.