axiom of infinity

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• Axiom of infinity — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Fraenkel axioms,… …   Wikipedia

• Inductive set (axiom of infinity) — In the context of the axiom of infinity, an inductive set (also known as a successor set) is a set X with the property that, for every x in X, the successor x of x is also an element of X. An example of an inductive set is the set of natural… …   Wikipedia

• Infinity — In mathematics, infinity is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: an infinite number of terms ) but it is a different type of number from the real numbers. Infinity is related to… …   Wikipedia

• Axiom of empty set — In set theory, the axiom of empty set is one of the axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads::exist x, forall… …   Wikipedia

• Axiom schema of replacement — In set theory, the axiom schema of replacement is a schema of axioms in Zermelo Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite… …   Wikipedia

• Axiom of regularity — In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo Fraenkel set theory and was introduced by harvtxt|von Neumann|1925. In first order logic the axiom reads::forall A (exists B (B in A)… …   Wikipedia

• Axiom of pairing — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Frankel axioms, the …   Wikipedia

• infinity, axiom of — Axiom needed in Russell s development of set theory to ensure that there are enough sets for the purposes of mathematics. In the theory of types, if there were only finitely many objects of the lowest type, then there would only be finite sets at …   Philosophy dictionary

• infinity — /in fin i tee/, n., pl. infinities. 1. the quality or state of being infinite. 2. something that is infinite. 3. infinite space, time, or quantity. 4. an infinite extent, amount, or number. 5. an indefinitely great amount or number. 6. Math. a.… …   Universalium

• Axiom of determinacy — The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two person games of length ω with perfect information. AD states that every such game in… …   Wikipedia

• Infinity-Borel set — In set theory, a subset of a Polish space X is infin; Borel if itcan be obtained by starting with the open subsets of X, and transfinitely iterating the operations of complementation and wellordered union (but see the caveat below). Formal… …   Wikipedia