• In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...
    36 KB (4,965 words) - 08:18, 29 March 2025
  • Thumbnail for Ordinal number
    In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...
    47 KB (6,689 words) - 14:57, 5 July 2025
  • an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal. The ordinals 1...
    2 KB (288 words) - 19:08, 18 July 2023
  • Epsilon number (category Ordinal numbers)
    numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation ε = ω ε , {\displaystyle...
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  • the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. The ordinals described here are not as large as...
    40 KB (5,519 words) - 23:29, 19 June 2025
  • Thumbnail for Cardinal number
    a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power...
    26 KB (3,834 words) - 18:49, 17 June 2025
  • interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory T {\displaystyle...
    52 KB (4,962 words) - 00:50, 20 June 2025
  • Thumbnail for Limit ordinal
    limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less...
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  • elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary...
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  • therefore, once again, not qualifying as an ordinal notation. Large countable ordinals Ordinal arithmetic Ordinal analysis Rathjen, Michael (1 August 2023)...
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  • Thumbnail for Natural number
    properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent...
    53 KB (5,887 words) - 07:23, 24 June 2025
  • the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called...
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  • counterexamples in topology. Epsilon numbers (mathematics) Large countable ordinal Ordinal arithmetic "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)"...
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  • finite ordinals). An ω-model is a model for a fragment of second-order arithmetic whose first-order part is the standard model of Peano arithmetic, but...
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  • Thumbnail for Georg Cantor
    of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact...
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  • called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor...
    15 KB (1,993 words) - 15:35, 7 February 2025
  • Thumbnail for Transfinite induction
    Transfinite induction (category Ordinal numbers)
    Transfinite number Well-founded induction Zorn's lemma J. Schlöder, Ordinal Arithmetic. Accessed 2022-03-24. It is not necessary here to assume separately...
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  • well-founded ordinals. Ordinal analysis was originated by Gentzen, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ε0....
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  • arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic,...
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  • Gentzen proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the ordinal called ε0 is wellfounded; see Gentzen's...
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  • smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank...
    21 KB (2,818 words) - 04:27, 23 June 2025
  • Thumbnail for Surreal number
    of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been...
    84 KB (11,658 words) - 08:57, 6 July 2025
  • the Guttman scale shown below in Table 2: Table 2. Data of the four ordinal arithmetic skill variables are hypothesized to form a Guttman scale The set profiles...
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  • best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement...
    38 KB (4,653 words) - 15:11, 22 June 2025
  • Thumbnail for Arithmetic
    Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider...
    165 KB (16,396 words) - 04:14, 2 June 2025
  • \omega ^{\omega }} are larger still. Arithmetic expressions containing ω {\displaystyle \omega } specify an ordinal number, and can be thought of as the...
    10 KB (1,232 words) - 08:58, 23 October 2024
  • Well-order (category Ordinal numbers)
    generalization Ordinal number Well-founded set Well partial order Prewellordering Directed set Manolios P, Vroon D. Algorithms for Ordinal Arithmetic. International...
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  • definition of the ordinals, and even a Δ 0 {\displaystyle \Delta _{0}} -formulation. Set induction in turn enables ordinal arithmetic in this sense. It...
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  • Thumbnail for Fundamental theorem of arithmetic
    In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every...
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  • Thumbnail for Set theory
    a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers...
    54 KB (6,586 words) - 11:37, 29 June 2025