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mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension 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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. 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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
.


Axioms

The set theory \mathrm^+_\infty of Olivier Esser consists of the following axioms:


Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equality (mathematics), equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned wi ...

\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y)


Positive comprehension

\exists x \forall y (y \in x \leftrightarrow \phi(y)) where \phi is a ''positive formula''. A positive formula uses only the logical constants \ but not \.


Closure

\exists x \forall y (y \in x \leftrightarrow \forall z (\forall w (\phi(w) \rightarrow w \in z) \rightarrow y \in z)) where \phi is a formula. That is, for every formula \phi, the intersection of all sets which contain every x such that \phi(x) exists. This is called the closure of \ 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 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.


Infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...

The von Neumann 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 In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally ...
.


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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally ...
).


See also

*
New Foundations In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Definition The well-formed fo ...
by
Quine Quine may refer to: * Quine (computing), a program that produces its source code as output * Quine's paradox, in logic * Quine (surname), people with the surname ** Willard Van Orman Quine (1908–2000), American philosopher and logician See al ...


References

*{{citation , last=Esser, first= Olivier , title=On the consistency of a positive theory. , journal=Mathematical Logic Quarterly, volume= 45 , year=1999, issue= 1, pages= 105–116 , mr=1669902 , doi=10.1002/malq.19990450110 Systems of set theory