Positive set theory

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 ${\displaystyle \phi }$ (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 ${\displaystyle GPK_{\infty }^{+}}$ of Olivier Esser consists of the following axioms:

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