Concept
Grand ordinal dénombrable
Concepts associés (8)
Ordinal analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0.
Théorème de Goodstein
En mathématiques, et plus précisément en logique mathématique, le 'théorème de Goodstein' est un énoncé arithmétique portant sur des suites, dites suites de Goodstein. Les suites de Goodstein sont des suites d'entiers à la croissance initiale extrêmement rapide, et le théorème établit que (en dépit des apparences) toute suite de Goodstein se termine par 0. Il doit son nom à son auteur, le mathématicien et logicien Reuben Goodstein.
Gentzen's consistency proof
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms.
Ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.
Hyperarithmetical theory
In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory. The central focus of hyperarithmetic theory is the sets of natural numbers known as hyperarithmetic sets. There are three equivalent ways of defining this class of sets; the study of the relationships between these different definitions is one motivation for the study of hyperarithmetical theory.
Nombre epsilon
En mathématiques, les nombres epsilon sont une collection de nombres transfinis définis par la propriété d'être des points fixes d'une application exponentielle. Ils ne peuvent donc pas être atteints à partir de 0 et d'un nombre fini d'exponentiations (et d'opérations « plus faibles », comme l'addition et la multiplication). La forme de base fut introduite par Georg Cantor dans le contexte du calcul sur les ordinaux comme étant les ordinaux ε satisfaisant l'équation où ω est le plus petit ordinal infini ; une extension aux nombres surréels a été découverte par John Horton Conway.
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 explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.
Nombre ordinal
vignette|Spirale représentant les nombres ordinaux inférieurs à ωω. En mathématiques, on appelle nombre ordinal un objet permettant de caractériser le type d'ordre d'un ensemble bien ordonné quelconque, tout comme en linguistique, les mots premier, deuxième, troisième, quatrième, etc. s'appellent des adjectifs numéraux ordinaux, et servent à préciser le rang d'un objet dans une collection, ou l'ordre d'un événement dans une succession.