Positive set theory

• " exists"

• The axiom of extensionality:.
• The axiom of empty set: there exists a set such that .
• 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 : 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 ; 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.

  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 certain topology on the classes.
• The theory can interpret ZFC. It in fact interprets a stronger theory.

  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.