**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

**Induction** — Most common meanings * Inductive reasoning, used in science and the scientific method * Mathematical induction, a method of proof in the field of mathematics * Electromagnetic induction in physics and engineering Other articles * Induction (play) … Wikipedia

**Mathematical induction** — can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers (positive… … Wikipedia

**Epsilon-induction** — In mathematics, in induction ( epsilon induction ) is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a given property P [ x ] . If the truth of the property for x follows from its truth for all… … 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

**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

**Large countable ordinal** — In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of… … Wikipedia

**Well-founded relation** — In mathematics, a binary relation, R, is well founded (or wellfounded) on a class X if and only if every non empty subset of X has a minimal element with respect to R; that is, for every non empty subset S of X, there is an element m of S such… … Wikipedia