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Positive set theory is the name of a certain group of axiomatic set theories originally created as an example of a (nonstandard) set theories in which the axiom of foundation fails (e.g. there exists $x$ such that $x\in x$). (Forti & Hinnion, 1989) Those theories are based on a weakening of the (inconsistent) comprehension axiom of naive set theory (which asserts that every formula $\phi(x)$ defines a set that contains all $x$ such that $\phi(x)$) by restraining the formulas used to a smaller class of formulas called positive formulas.
While most positive set theories are weaker than $\text{ZFC}$ (and usually mutually interpretable with second-order arithmetic), one of them, $\text{GPK}^+_\infty$ turns out to be very powerful, being mutually interpretable with Morse-Kelley set theory plus an axiom asserting that the class of all ordinals is weakly compact. (Esser, 1997)
In the first-order language $\{=,\in\}$, we define a BPF formula (bounded positive formula) the following way (Esser, 1997): For every variable $x$, $y$ and BPF formulas $\varphi$, $\psi$,
A formula is then a GPF formula (generalized positive formula) if it is a BPF formula or if it is of the form $\forall x(\theta(x)\Rightarrow\varphi)$ with $\theta(x)$ a GPF formula with exactly one free variable $x$ and no parameter and $\varphi$ is a GPF formula (possibly with parameters). (Esser, 1996)
The positive set theory $\text{GPK}$ consists of the following axioms:
The empty set axiom is necessary, as without it the theory would hold in the trivial model which has only one element satisfying $x=\{x\}$. Note that, while $\text{GPK}$ do proves the existence of a set such that $x\in x$, Olivier Esser proved that it refutes the anti-foundation axiom (AFA). (Esser, 1996)
The theory $\text{GPK}^+$ is obtained by adding the following axiom:
This axiom scheme asserts that for any (possibly proper) class $C=\{x|\varphi(x)\}$ there is a smallest set $X$ containing $C$, i.e. $C\subset X$ and for all sets $Y$ such that $C\subset Y$, one has $X\subset Y$. (Esser, 1999)
Note that replacing GPF comprehension in $\text{GPK}^+$ by BPF comprehension does not make the theory any weaker: BPF comprehension plus Closure implies GPF comprehension.
Both $\text{GPK}$ and $\text{GPK}^+$ are consistent relative to $\text{ZFC}$, in fact mutually interpretable with second-order arithmetic. However a much stronger theory, $\text{GPK}^+_\infty$, is obtained by adding the following axiom:
By “von Neumann ordinal” we mean the usual definition of ordinals as well-ordered-by-inclusion sets containing all the smaller ordinals. Here $\omega$ is the set of all finite ordinals (the natural numbers). The point of this axiom is not implying the existence of an infinite set; the class $\omega$ exists, so it has a set closure which is certainely infinite. This set closure happens to satisfy the usual axiom of infinity of $\text{ZFC}$ (i.e. it contains 0 and the successor of all its members) but in $\text{GPK}^+$ this is not enough to deduce that $\omega$ itself is a set (an improper class).
Olivier Esser showed that $\text{GPK}^+_\infty$ can not only interpret $\text{ZFC}$ (and prove it consistent), but is in fact mutually interpretable with a much stronger set theory, namely, Morse-Kelley set theory with an axiom asserting that the (proper) class of all ordinals is weakly compact. This theory is powerful enough to prove, for instance, that there exists a proper class of hyper-Mahlo cardinals. (Esser, 1997)
To be expanded.
To be expanded. (Esser, 2000; Forti & Hinnion, 1989)
To be expanded. (Esser, 2003)
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