*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 $x\in x$). {% cite Forti1989 %} Those theories are based on a weakening of the (inconsistent) *comprehension axiom* of <a href="index.php?title=Naive_set_theory&amp;action=edit&amp;redlink=1" class="new" title="Naive set theory (page does not exist)">naive set theory</a> (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 [$\text{ZFC}$](ZFC.md "ZFC") (and usually mutually interpretable with <a href="http://en.wikipedia.org/wiki/second-order_arithmetic" class="extiw" title="wikipedia:second-order arithmetic">second-order arithmetic</a>), one of them, $\text{GPK}^+_\infty$ turns out to be very powerful, being mutually interpretable with <a href="Morse-Kelley_set_theory" class="mw-redirect" title="Morse-Kelley set theory">Morse-Kelley set theory</a> plus an axiom asserting that the class of all [ordinals](Ordinal.md "Ordinal") is [[weakly compact]]. {% cite Esser1997 %} ## Positive formulas In the first-order language $\{=,\in\}$, we define a *BPF formula* (bounded positive formula) the following way {% cite Esser1997 %}: For every variable $x$, $y$ and BPF formulas $\varphi$, $\psi$, - $x=y$ and $x\in y$ are BPF. - $\varphi\land\psi$, $\varphi\lor\psi$, $\exists x\varphi$ and $(\forall x\in y)\varphi$ are BPF. 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). {% cite Esser1996 %} ## $\text{GPK}$ positive set theories The positive set theory $\text{GPK}$ consists of the following axioms: - **Empty set**: $\exists x\forall y(y\not\in x)$. - **Extensionality**: $\forall x\forall y(x=y\Leftrightarrow\forall z(z\in x\Leftrightarrow z\in y))$. - **GPF comprehension**: the universal closure of $\exists x\forall y(y\in x\Leftrightarrow\varphi)$ for every GPF formula $\varphi$ (with parameters) in which $x$ does not occur. 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 <a href="http://en.wikipedia.org/wiki/anti-foundation_axiom" class="extiw" title="wikipedia:anti-foundation axiom">anti-foundation axiom</a> (AFA). {% cite Esser1996 %} The theory $\text{GPK}^+$ is obtained by adding the following axiom: - **Closure**: the universal closure of $\exists x(\forall z(\varphi(z)\Rightarrow z\in x)\land\forall y(\forall z(\varphi(z)\Rightarrow z\in y)\Rightarrow x\subset y))$ for every formula $\varphi(z)$ (not necessarily BPF or GPF) with a free variable $z$ (and possibly parameters) such that $x$ does not occur in $\varphi$. 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 $X\subset Y$. {% cite Esser1999 %} 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: - **Infinity**: the von Neumann ordinal $\omega$ is a set. 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](Mahlo.md "Mahlo") cardinals. {% cite Esser1997 %} ## As a topological set theory *To be expanded.* ## The axiom of choice and $\text{GPK}$ set theories *To be expanded. {% cite Esser2000 Forti1989 %}* ## Other positive set theories and the inconsistency of the axiom of extensionality *To be expanded. {% cite Esser2003 %}* This article is a stub. Please help us to improve Cantor's Attic by adding information.