In mathematical logic, positive set theory is an alternative set theory consisting of the following axioms: Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

• The axiom of extensionality: .
• The axiom of infinity: the von Neumann ordinal ω exists.
• The axiom of closure: for every set x, a set exists which is the intersection of all sets containing x; this is called the closure of x and is written {x}.
• The axiom of empty set: there exists a set such that .
• The axiom of comprehension: if φ is a formula in predicate logic using only , , , , = , and , then the set of all x such that φ(x) is also a set. Quantification (, ) may be bounded.
  • Note that negation is specifically not permitted.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ... A separate article covers Saint John Neumann, the American priest. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ...

Interesting properties

• The universal set is a proper set in this theory.
• The theory can interpret ZFC (by restricting oneself to the set of sets whose complement is also a set).
• The set of all well-founded sets is a proper set.

In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

Researchers

Oliver Esser seems to be the most active in this field.

Related

See also Quine's New Foundations W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... In mathematical logic, the New Foundations (NF) of W. V. O. Quine is a candidate set theory, obtained from a streamlined version of the theory of types of Bertrand Russell. ...