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Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic.
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
2. Zermelo−Fraenkel set theory is the standard system of axioms for set theory 3. Zermelo set theory is similar to the usual Zermelo-Fraenkel set theory, but without the axioms of replacement and foundation 4. Zermelo's well-ordering theorem states that every set can be well ordered ZF Zermelo−Fraenkel set theory without the axiom of choice ZFA
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set.
The axiom schema of specification is characteristic of systems of axiomatic set theory related to the usual set theory ZFC, but does not usually appear in radically different systems of alternative set theory. For example, New Foundations and positive set theory use different restrictions of the axiom of comprehension of naive set theory.
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.
Naive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent due to Russell's paradox. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker axiom schema of separation .