# Zermelo-Fraenkel set theory

Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) are systems of axiomatic set theories free of paradoxes. The latter includes the infamous axiom of choice.

ZFC with the axiom of choice (C stands for "choice") is a system upon which the whole of mathematics can be based (but it's not the only pretender for that title). The axiom of choice is independent of other axioms of ZF set theory.

The founders, mathematicians Ernst Zermelo and Abraham Fraenkel, proposed the theory in the early XX century, basing it on a system of Aristotelian logic and ZF axioms, including the controversial Axiom of Choice.

ZFC axioms: 1. Axiom of Extensionality 2. Axiom of Regularity 3. Axiom Schema of Specification 4. Axiom of Pairing 5. Axiom of Union 6. Axiom Schema Of Replacement 7. Axiom of Infinity 8. Axiom of Power Set 9. Well-ordering theorem 10. Axiom of Choice

* **Universal set** is a set containing all sets.
* **Hereditary set** or a **pure set** is a set whose elements are all hereditary sets, i.e. all elements of the set are themselves sets, as are all elements of the elements, and so on.
* **Ur-element** is a concrete or abstract object that is not a set, but may be an element of a set.
* **Proper class** is a collection of mathematical objects defined by a property shared by its elements which are too big to be sets.

ZFC intended to formalize a notion of **hereditary well-founded set** so that all entities in the universe of discourse are such sets.

Thus, ZFC axioms refer only to pure sets and disallow prevent its models from containing _urelements_. Furthermore, proper classes can only be treated indirectly.

Specifically, ZFC does not allow for the existence of a universal set, nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory is a commonly used conservative extension of ZFC that does allow explicit treatment of proper classes.

ZFC is a _one-sorted theory_ in _first-order logic_. The signature has equality and a single primitive binary relation: set membership denoted as `∈`.

There are many equivalent formulations of the ZFC axioms. Most of the ZFC axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axioms describe properties of set membership.

A goal of the ZFC 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). The metamathematics of ZFC has been extensively studied. Landmark results in this area established the independence of the continuum hypothesis from ZFC, and of the axiom of choice from the remaining ZFC axioms. The consistency of a theory such as ZFC cannot be proved within the theory itself.

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