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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.
Additionally, if one takes ZFC set theory (i.e., ZF with the axiom of choice), removes the axiom of replacement and the axiom of collection, but keeps the axiom schema of specification, one gets the weaker system of axioms called ZC (i.e., Zermelo's axioms, plus the axiom of choice). [11]
The axiom of extensionality, [1] [2] also called the axiom of extent, [3] [4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. [5] [6] The axiom defines what a set is. [1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.
This category is for axioms in the language of set theory; roughly speaking, ones that "talk about sets". Inclusion in this category does not necessarily imply that the axiom in question is an accepted axiom, or that it is believed to be true in the von Neumann universe of sets.