Positive Set Theory

Set Theories in which the Axiom Of Comprehension

" $\{x \mid \phi\}$ exists"

holds for at least the positive formulas

\phi

(the smallest class of formulas
containing atomic membership and equality formulas and closed under conjunction, disjunction,
existential and universal quantification).

Typically, the motivation for these theories is topological: the sets are the classes which
are closed under a certain Topology . The closure conditions for the various constructions
allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the
existential quantifier seems to require that the topology be Compact .

The set theory

GPK^+_{\infty}

of Olivier Esser consists of the following axioms:

The Axiom Of Extensionality : $x=y \Leftrightarroworall a\, (a\in x \Leftrightarrow a\in y)$ .

The Axiom Of Empty Set : there exists a set $\emptyset$ such that

\perp

The axiom of generalized positive Comprehension : if $\phi$ is a formula in predicate logic using only $ee$ , $\wedge$ , $\exists$ , $orall$ , $=$ , and $\in$ , then the set of all $x$ such that $\phi(x)$ is also a set. Quantification ( $orall$ , $\exists$ ) may be bounded.

--- Note that negation is specifically not permitted.

The axiom of ): for any class ''C'' there is a set which is the intersection of all sets which contain ''C'' as a subclass. This is obviously a reasonable principle if the sets are understood as closed classes in a topology.

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

INTERESTING PROPERTIES

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 ).

... many more -->

RESEARCHERS

R. J. 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.

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	CATEGORIES ABOUT POSITIVE SET THEORY

	systems of set theory

Information About

Positive Set Theory