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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", [1] and ZF refers to the ...
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 ...
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A.
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 power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. [2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set ...