The set theory ${\displaystyle \mathrm {GPK} _{\infty }^{+}}$  of Olivier Esser consists of the following axioms:^[1]

Extensionality edit

${\displaystyle \forall x\forall y(\forall z(z\in x\leftrightarrow z\in y)\to x=y)}$ 

Positive comprehension edit

${\displaystyle \exists x\forall y(y\in x\leftrightarrow \phi (y))}$ 

where ${\displaystyle \phi }$  is a positive formula. A positive formula uses only the logical constants ${\displaystyle \{\top ,\bot ,\land ,\lor ,\forall ,\exists ,=,\in \}}$  but not ${\displaystyle \{\to ,\neg \}}$ .

Closure edit

${\displaystyle \exists x\forall y(y\in x\leftrightarrow \forall z(\forall w(\phi (w)\rightarrow w\in z)\rightarrow y\in z))}$ 

where ${\displaystyle \phi }$  is a formula. That is, for every formula ${\displaystyle \phi }$ , the intersection of all sets which contain every ${\displaystyle x}$  such that ${\displaystyle \phi (x)}$  exists. This is called the closure of ${\displaystyle \{x\mid \phi (x)\}}$  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.

Infinity edit

The von Neumann ordinal ${\displaystyle \omega }$  exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of ${\displaystyle \omega }$  exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that ${\displaystyle \omega }$  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.

• 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 (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory (Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal).

• 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.^{[citation needed]}

• New Foundations by Quine

• Esser, Olivier (1999), "On the consistency of a positive theory.", Mathematical Logic Quarterly, 45 (1): 105–116, doi:10.1002/malq.19990450110, MR 1669902