. . . . . "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 relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available. Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal \u03C91; their supremum is called Church\u2013Kleene \u03C91 or \u03C91CK (not to be confused with the first uncountable ordinal, \u03C91), described . Ordinal numbers below \u03C91CK are the recursive ordinals (see ). Countable ordinals larger than this may still be defined, but do not have notations. Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. The ordinals described here are not as large as the ones described in large cardinals, but they are large among those that have constructive notations (descriptions). Larger and larger ordinals can be defined, but they become more and more difficult to describe."@en . . . . . . . . . . . . . . . . "Na disciplina de teoria dos conjuntos, h\u00E1 muitas maneiras de descrever ordinais espec\u00EDficos eque s\u00E3o cont\u00E1veis. As menores descri\u00E7\u00F5es podem ser claramente e de forma n\u00E3o-redundante expressas em termos de suas formas normais Cantor. Al\u00E9m disso, muitos ordinais relevantes para a teoria da prova ainda tem nota\u00E7\u00F5es ordinais comput\u00E1veis. No entanto, n\u00E3o \u00E9 poss\u00EDvel decidir eficazmente se uma determinada nota\u00E7\u00E3o ordinal \u00E9 nota\u00E7\u00E3o ou n\u00E3o (por raz\u00F5es de certa forma an\u00E1logas, \u00E0 insolubilidade do problema da parada); existem v\u00E1rias maneiras de saber se uma nota\u00E7\u00E3o ordinal \u00E9 defin\u00EDvel."@pt . . . . . . "4376814"^^ . . . . . . . . . . . . . "En math\u00E9matiques, et plus particuli\u00E8rement en th\u00E9orie des ensembles, il existe de nombreuses m\u00E9thodes de description des ordinaux d\u00E9nombrables. Les plus petits (jusqu'\u00E0 \u03B50) peuvent \u00EAtre exprim\u00E9s (de fa\u00E7on utile et non circulaire) \u00E0 l'aide de leur forme normale de Cantor. Au-del\u00E0, on parle de grands ordinaux d\u00E9nombrables ; de nombreux grands ordinaux (le plus souvent en rapport avec la th\u00E9orie de la d\u00E9monstration) poss\u00E8dent des notations ordinales calculables. Cependant, il n'est pas possible en g\u00E9n\u00E9ral de d\u00E9cider si une notation ordinale potentielle en est effectivement une, pour des raisons analogues \u00E0 celles rendant insoluble le probl\u00E8me de l'arr\u00EAt. Comme il ne peut exister qu'un nombre d\u00E9nombrable de notations, l'ensemble des ordinaux qui en admettent une se termine bien avant le premier ordinal non d\u00E9nombrable \u03C91 ; la borne sup\u00E9rieure de cet ensemble s'appelle l'ordinal \u03C91 de Church\u2013Kleene, not\u00E9 \u03C91CK (cet ordinal est d\u00E9nombrable, et ne doit pas \u00EAtre confondu avec \u03C91). Les ordinaux inf\u00E9rieurs \u00E0 \u03C91CK sont les ordinaux r\u00E9cursifs. Il est possible de d\u00E9finir des ordinaux sup\u00E9rieurs, mais ils ne poss\u00E8deront pas de notations. L'\u00E9tude des ordinaux d\u00E9nombrables non r\u00E9cursifs est d\u00E9licate, la difficult\u00E9 principale venant de ce qu'on ne sait pas, en g\u00E9n\u00E9ral, comparer deux grands ordinaux d\u00E9finis par des m\u00E9thodes diff\u00E9rentes, et parfois m\u00EAme, qu'on ne sait pas d\u00E9montrer qu'un ordre donn\u00E9 est un bon ordre."@fr . "Grand ordinal d\u00E9nombrable"@fr . . . . . . . . . "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 relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available."@en . . "En math\u00E9matiques, et plus particuli\u00E8rement en th\u00E9orie des ensembles, il existe de nombreuses m\u00E9thodes de description des ordinaux d\u00E9nombrables. Les plus petits (jusqu'\u00E0 \u03B50) peuvent \u00EAtre exprim\u00E9s (de fa\u00E7on utile et non circulaire) \u00E0 l'aide de leur forme normale de Cantor. Au-del\u00E0, on parle de grands ordinaux d\u00E9nombrables ; de nombreux grands ordinaux (le plus souvent en rapport avec la th\u00E9orie de la d\u00E9monstration) poss\u00E8dent des notations ordinales calculables. Cependant, il n'est pas possible en g\u00E9n\u00E9ral de d\u00E9cider si une notation ordinale potentielle en est effectivement une, pour des raisons analogues \u00E0 celles rendant insoluble le probl\u00E8me de l'arr\u00EAt."@fr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "1123338159"^^ . . . . . . . "Na disciplina de teoria dos conjuntos, h\u00E1 muitas maneiras de descrever ordinais espec\u00EDficos eque s\u00E3o cont\u00E1veis. As menores descri\u00E7\u00F5es podem ser claramente e de forma n\u00E3o-redundante expressas em termos de suas formas normais Cantor. Al\u00E9m disso, muitos ordinais relevantes para a teoria da prova ainda tem nota\u00E7\u00F5es ordinais comput\u00E1veis. No entanto, n\u00E3o \u00E9 poss\u00EDvel decidir eficazmente se uma determinada nota\u00E7\u00E3o ordinal \u00E9 nota\u00E7\u00E3o ou n\u00E3o (por raz\u00F5es de certa forma an\u00E1logas, \u00E0 insolubilidade do problema da parada); existem v\u00E1rias maneiras de saber se uma nota\u00E7\u00E3o ordinal \u00E9 defin\u00EDvel. Desde que existam muitas nota\u00E7\u00F5es cont\u00E1veis, todos os ordinal com nota\u00E7\u00F5es s\u00E3o completamente explorados sob o primeiro o ordinal n\u00E3o cont\u00E1vel w1; seu supremo \u00E9 chamado Church-Kleene w1 ou w1ck (n\u00E3o confunda com o primeiro ordinal n\u00E3o-cont\u00E1vel, w1), descrito abaixo. N\u00FAmeros ordinais abaixo de w1ck s\u00E3o ordinais recursivos (veja abaixo). Ordinais cont\u00E1veis maiores do que este pode ainda ser definida, mas n\u00E3o t\u00EAm nota\u00E7\u00F5es. Devido ao foco nos ordinais cont\u00E1veis, a aritm\u00E9tica ordinal \u00E9 usada por toda parte, exceto onde o contr\u00E1rio \u00E9 indicado. Os ordinais descritos aqui n\u00E3o s\u00E3o t\u00E3o grandes como os cardinais longos, mas eles s\u00E3o grandes entre aqueles que t\u00EAm nota\u00E7\u00F5es construtivas (descri\u00E7\u00F5es). Ordinais cada vez maiores pode ser definida, mas tornam-se mais e mais dif\u00EDcil de descrever."@pt . . . . . . . "35751"^^ . . . . "Grandes ordinais cont\u00E1veis"@pt . . "Large countable ordinal"@en . . . . . . . . . . .