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transfinite order

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  • Transfinite induction — is an extension of mathematical induction to well ordered sets, for instance to sets of ordinals or cardinals. Transfinite induction Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is …   Wikipedia

  • Order topology — In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order …   Wikipedia

  • List of first-order theories — In mathematical logic, a first order theory is given by a set of axioms in somelanguage. This entry lists some of the more common examples used in model theory and some of their properties. PreliminariesFor every natural mathematical structure… …   Wikipedia

  • Well-order — In mathematics, a well order relation (or well ordering) on a set S is a total order on S with the property that every non empty subset of S has a least element in this ordering.Equivalently, a well ordering is a well founded total order.The set… …   Wikipedia

  • Completeness (order theory) — In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). A special use of the term refers to complete partial orders or complete lattices.… …   Wikipedia

  • Ordinal number — This article is about the mathematical concept. For number words denoting a position in a sequence ( first , second , third , etc.), see Ordinal number (linguistics). Representation of the ordinal numbers up to ωω. Each turn of the spiral… …   Wikipedia

  • Surreal number — In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share… …   Wikipedia

  • set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… …   Universalium

  • Reverse mathematics — is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. The method can briefly be described as going backwards from the theorems to the axioms. This contrasts with the ordinary… …   Wikipedia

  • Georg Cantor — Infobox Scientist name = Georg Ferdinand Ludwig Cantor image width=225px caption = birth date = birth date|1845|3|3 birth place = Saint Petersburg, Russia death date = death date and age|1918|1|6|1845|3|3 death place = Halle, Germany residence =… …   Wikipedia

  • Gentzen's consistency proof — Gentzen s theoremIn 1936 Gerhard Gentzen proved the consistency of first order arithmetic using combinatorial methods. Gentzen s proof shows much more than merely that first order arithmetic is consistent. Gentzen showed that the consistency of… …   Wikipedia

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