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Ordinal arithmetic

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  • Ordinal arithmetic — In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an… …   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

  • Ordinal notation — In mathematical logic and set theory, an ordinal notation is a finite sequence of symbols from a finite alphabet which names an ordinal number according to some scheme which gives meaning to the language. There are many such schemes of ordinal… …   Wikipedia

  • Ordinal analysis — In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that… …   Wikipedia

  • Ordinal collapsing function — In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger …   Wikipedia

  • ordinal variable — noun A variable with values whose order is significant, but on which no meaningful arithmetic like operations can be performed …   Wiktionary

  • 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

  • Limit ordinal — A limit ordinal is an ordinal number which is neither zero nor a successor ordinal. Various equivalent ways to express this are: *It cannot be reached via the ordinal successor operation S ; in precise terms, we say lambda; is a limit ordinal if… …   Wikipedia

  • Successor ordinal — When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. Using von Neumann s ordinal numbers (the standard ordinals used in set theory), we have, for …   Wikipedia

  • Additively indecomposable ordinal — In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any eta,gamma …   Wikipedia

  • Transfinite arithmetic — In mathematics, transfinite arithmetic is the generalization of elementary arithmetic to infinite quantities like infinite sets. It was originally invented by the German mathematician Georg Cantor. See also * transfinite number * cardinal… …   Wikipedia