Editing Positive set theory (section)

== Axioms ==

The set theory <math>\mathrm{GPK}^+_\infty</math> of Olivier Esser consists of the following axioms:<ref>{{Cite SEP|url-id=settheory-alternative|title=Alternative Axiomatic Set Theories|first=M. Randall|last=Holmes|date=21 September 2021}}</ref>

=== [[axiom of extensionality|Extensionality]] ===

<math>\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y)</math>

=== Positive [[axiom of comprehension|comprehension]] ===

<math>\exists x \forall y (y \in x \leftrightarrow \phi(y))</math>

where <math>\phi</math> is a ''positive formula''. A positive formula uses only the [[logical constants]] <math>\{\top, \bot, \land, \lor, \forall, \exists, =, \in\}</math> but not <math>\{\to, \neg\}</math>.

=== [[topological closure|Closure]] ===

<math>\exists x \forall y (y \in x \leftrightarrow \forall z (\forall w (\phi(w) \rightarrow w \in z) \rightarrow y \in z))</math>

where <math>\phi</math> is a formula. That is, for every formula <math>\phi</math>, the intersection of all sets which contain every <math>x</math> such that <math>\phi(x)</math> exists. This is called the closure of <math>\{x \mid \phi(x)\}</math> 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 [[Von Neumann–Bernays–Gödel set theory|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.

=== [[axiom of infinity|Infinity]] ===

The [[John von Neumann|von Neumann]] [[ordinal number|ordinal]] <math>\omega</math> exists.  This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of <math>\omega</math> exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that <math>\omega</math> 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]].