positive set theory

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

The axiom of extensionality ( $x=y \Leftrightarrow (a\in x \Leftrightarrow a\in y)$ )
The axiom of infinity (the Von Neumann Ordinals $\omega$ exists)
The axiom of closure for every set $x$ , a set exists which is the intersection of all sets containing $x$ ; this is called the closure of x and is written $\{x\}$
The axiom of empty set (there exists a set $\emptyset$ such that $\neg \exists_x x\in\emptyset$ )
The axiom of comprehension If $\phi$ is a formula in propositional logic using only $\vee$ , $\wedge$ , $\exists$ , $\forall$ , $=$ , and $\in$ , then the set of all $x$ such that $\phi(x)$ is also a set.

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.

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