Zermelo Frankel set theory - meaning and definition. What is Zermelo Frankel set theory
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What (who) is Zermelo Frankel set theory - definition

STANDARD FORM OF AXIOMATIC SET THEORY
Zermelo-Fraenkel axiom; ZFC; Zermelo-Fraenkel axioms; Zermelo-Frankel axioms; Zermelo-Frankel set theory; ZFC set theory; Zermelo-Fraenkel framework; ZFC Set Theory; Zermelo-Fränkel set theory; Zermelo-Frankel; Zfc; ZFC set; ZF axioms; Zermelo–Frankel set theory; Zermelo-Fraenkel set theory; ZF set theory; Zermelo-Fraenkel-Skolem set theory; Zermelo-Frankel axiom; Zermelo–Fraenkel axioms; Zermelo Fraenkel set theory; Zermelo-Fraenkel; Zermelo–Fraenkel axiomatization; Zermelo-Fraenkel axiomatization; Zermelo-Fränkel; ZFC Set theory; Zermelo–Fraenkel axiom; Zermelo–Fraenkel framework; Zermelo–Fraenkel; Zermelo-frankel; Zermelo–Fraenkel set theory with the axiom of choice; Axioms of ZF; Zermelo−Fraenkel set theory; Zermelo-Fraenkel set theory with the axiom of choice

Zermelo Frankel set theory      
<mathematics> A set theory with the axioms of {Zermelo set theory} (Extensionality, Union, Pair-set, Foundation, Restriction, Infinity, Power-set) plus the Replacement {axiom schema}: If F(x,y) is a formula such that for any x, there is a unique y making F true, and X is a set, then F x : x in X is a set. In other words, if you do something to each element of a set, the result is a set. An important but controversial axiom which is NOT part of ZF theory is the Axiom of Choice. (1995-04-10)
Zermelo–Fraenkel set theory         
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.
ZFC         
<mathematics> Zermelo Frankel set theory plus the {Axiom of Choice}. A favourite axiomatisation of set theory. (1995-03-29)

Wikipedia

Zermelo–Fraenkel set theory

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. 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. For example, the axiom of pairing says that given any two sets a {\displaystyle a} and b {\displaystyle b} there is a new set { a , b } {\displaystyle \{a,b\}} containing exactly a {\displaystyle a} and b {\displaystyle b} . Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, intended to formalize set membership, which is usually denoted {\displaystyle \in } . The formula a b {\displaystyle a\in b} means that the set a {\displaystyle a} is a member of the set b {\displaystyle b} (which is also read, " a {\displaystyle a} is an element of b {\displaystyle b} " or " a {\displaystyle a} is in b {\displaystyle b} ").

The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see Axiom of choice § Independence) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.