Positive set theory

• Positive formulas
• $\text{GPK}$ positive set theories
• As a topological set theory
• The axiom of choice and $\text{GPK}$ set theories
• Other positive set theories and the inconsistency of the axiom of extensionality

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)

Positive formulas

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 $\varphi$, $\psi$,

• $x=y$ 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). (Esser, 1996)

$\text{GPK}$ 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 anti-foundation axiom (AFA). (Esser, 1996)

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 $X\subset Y$. (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:

• 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 cardinals. (Esser, 1997)

As a topological set theory

To be expanded.

The axiom of choice and $\text{GPK}$ set theories

To be expanded. (Esser, 2000; Forti & Hinnion, 1989)

Other positive set theories and the inconsistency of the axiom of extensionality

To be expanded. (Esser, 2003)

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References

1. Forti, M., & Hinnion, R. (1989). The Consistency Problem for Positive Comprehension Principles. J. Symbolic Logic, 54(4), 1401–1418.
2. Esser, O. (1997). An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory. Mathematical Logic Quarterly, 43, 369–377. https://doi.org/10.1002/malq.19970430309
3. Esser, O. (1996). Inconsistency of GPK+AFA. Mathematical Logic Quarterly, 42, 104–108. https://doi.org/10.1002/malq.19960420109
4. Esser, O. (1999). On the Consistency of a Positive Theory. Mathematical Logic Quarterly, 45, 105–116. https://doi.org/10.1002/malq.19990450110
5. Esser, O. (2000). Inconsistency of the Axiom of Choice with the Positive Theory GPK^+_∞. Journal of Symbolic Logic, 65(4), 1911–1916. https://doi.org/10.2307/2695086
6. Esser, O. (2003). On the axiom of extensionality in the positive set theory. Mathematical Logic Quarterly, 19, 97–100. https://doi.org/10.1002/malq.200310009

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