Positive set theory

In mathematical logic, positive set theory is an alternative set theory consisting of the following axioms:

The axiom of extensionality (<math>x=y \Leftrightarrow (a\in x \Leftrightarrow a\in y)<math>)
The axiom of infinity (the Von Neumann Ordinals <math>\omega<math> exists)
The axiom of closure for every set <math>x<math>, a set exists which is the intersection of all sets containing <math>x<math>; this is called the closure of x and is written <math>\{x\}<math>
The axiom of empty set (there exists a set <math>\emptyset<math> such that <math>\neg \exists_x x\in\emptyset<math>)
The axiom of comprehension If <math>\phi<math> is a formula in propositional logic using only <math>\vee<math>, <math>\wedge<math>, <math>\exists<math>, <math>\forall<math>, <math>=<math>, and <math>\in<math>, then the set of all <math>x<math> such that <math>\phi(x)<math> is also a set.
- Note that negation is specifically not permitted
- Quantification (<math>\forall<math>, <math>\exists<math>) may be bounded

Interesting properties

The universal set is a proper set in this theory
The theory can interpret ZFC (by restricting oneself to the set of sets whose complement is also a set)
The set of all well-founded sets is a proper set

Oliver Esser seems to be the most active in this field.