. "Em matem\u00E1tica, especialmente em teoria de conjuntos, um cardinal \u00E9 denominado regular se ele \u00E9 igual a sua pr\u00F3pria cofinalidade. Caso contr\u00E1rio, \u00E9 dito singular."@pt . . . . . . . . "En th\u00E9orie des ensembles, un cardinal infini est dit r\u00E9gulier s'il est \u00E9gal \u00E0 sa cofinalit\u00E9. Intuitivement, un cardinal est r\u00E9gulier si toute r\u00E9union index\u00E9e par un ensemble petit d'ensembles petits est petite, o\u00F9 un ensemble est dit petit s'il est de cardinalit\u00E9 strictement inf\u00E9rieure \u00E0 . Une autre d\u00E9finition possible \u00E9quivalente est que est r\u00E9gulier si pour tout cardinal , toute fonction est born\u00E9e. Un cardinal qui n'est pas r\u00E9gulier est dit singulier."@fr . . . . . . "1108320247"^^ . "En th\u00E9orie des ensembles, un cardinal infini est dit r\u00E9gulier s'il est \u00E9gal \u00E0 sa cofinalit\u00E9. Intuitivement, un cardinal est r\u00E9gulier si toute r\u00E9union index\u00E9e par un ensemble petit d'ensembles petits est petite, o\u00F9 un ensemble est dit petit s'il est de cardinalit\u00E9 strictement inf\u00E9rieure \u00E0 . Une autre d\u00E9finition possible \u00E9quivalente est que est r\u00E9gulier si pour tout cardinal , toute fonction est born\u00E9e. Un cardinal qui n'est pas r\u00E9gulier est dit singulier. Par exemple, pour , petit signifie fini, or toute r\u00E9union index\u00E9e par un ensemble fini d'ensembles finis est finie, donc est un cardinal r\u00E9gulier. Pour , petit signifie d\u00E9nombrable, or, sous l'axiome du choix d\u00E9nombrable, toute r\u00E9union index\u00E9e par un ensemble d\u00E9nombrable d'ensembles d\u00E9nombrables est d\u00E9nombrable, donc est r\u00E9gulier. On peut montrer, sous l'axiome du choix, qu'il en est de m\u00EAme pour tout cardinal successeur : si est un ordinal, alors est r\u00E9gulier. C'est une cons\u00E9quence simple du fait que . Un cardinal singulier est n\u00E9cessairement un cardinal limite. Une question naturelle se pose : la r\u00E9ciproque est-elle vraie ? Un contre-exemple \u00E0 cette r\u00E9ciproque, c'est-\u00E0-dire un cardinal limite et r\u00E9gulier, est appel\u00E9 cardinal faiblement inaccessible. Le premier cardinal singulier est . En effet, , il peut donc s'\u00E9crire comme une r\u00E9union index\u00E9e par un ensemble d\u00E9nombrable d'ensembles de cardinalit\u00E9 strictement inf\u00E9rieure."@fr . . "\u96C6\u5408\u8AD6\u306B\u304A\u3044\u3066\u3001\u6B63\u5247\u57FA\u6570\uFF08\u305B\u3044\u305D\u304F\u304D\u3059\u3046\u3001\u82F1: regular cardinal\uFF09\u3068\u306F\u3001\u305D\u306E\u5171\u7D42\u6570\u304C\u305D\u308C\u81EA\u8EAB\u3067\u3042\u308B\u57FA\u6570\u306E\u3053\u3068\u3002 \u7C21\u5358\u306B\u8A00\u3048\u3070\u3001\u6B63\u5247\u57FA\u6570\u306F\u5C0F\u3055\u3044\u30D1\u30FC\u30C4\u306E\u5C11\u306A\u3044\u96C6\u307E\u308A\u306B\u5206\u5272\u3067\u304D\u306A\u3044\u3082\u306E\u3067\u3042\u308B\u3002 (\u3053\u306E\u72B6\u6CC1\u306F\u9078\u629E\u516C\u7406\u3092\u4EEE\u5B9A\u3057\u306A\u3044\u6587\u8108\u3067\u306F\u3082\u3063\u3068\u8907\u96D1\u3067\u3042\u308B\u3002\u305D\u306E\u3088\u3046\u306A\u5834\u5408\u3001\u5168\u3066\u306E\u6FC3\u5EA6\u304C\u6574\u5217\u96C6\u5408\u306E\u6FC3\u5EA6\u3068\u306F\u9650\u3089\u306A\u304F\u3001\u4E0A\u8A18\u306E\u5B9A\u7FA9\u306F\u6574\u5217\u96C6\u5408\u306E\u6FC3\u5EA6\u306E\u307F\u306B\u5BFE\u3057\u3066\u306A\u3055\u308C\u308B\u3002) \u9078\u629E\u516C\u7406\u3092\u4EEE\u5B9A\u3059\u308B\u3068\u304D\u306F\u3001\u3044\u304B\u306A\u308B\u6FC3\u5EA6\u3082\u57FA\u6570\u306B\u306A\u308A\u3001\u7121\u9650\u57FA\u6570 \u304C\u6B63\u5247\u3067\u3042\u308B\u3053\u3068\u306F \u672A\u6E80\u306E\u57FA\u6570\u306E \u672A\u6E80\u500B\u306E\u548C\u3067\u306F\u8868\u305B\u306A\u3044\u3053\u3068\u3068\u540C\u5024\u306B\u306A\u308B\u3002 \u307E\u305F\u3001\u7121\u9650\u9806\u5E8F\u6570 \u304C\u6B63\u5247\u9806\u5E8F\u6570\u3068\u547C\u3070\u308C\u308B\u306E\u306F\u3001\u305D\u308C\u304C\u6975\u9650\u9806\u5E8F\u6570\u3067\u3088\u308A\u5C0F\u3055\u3044\u9806\u5E8F\u6570\u306E\u9806\u5E8F\u578B\u304C \u672A\u6E80\u306E\u96C6\u5408\u306E\u6975\u9650\u306B\u306A\u3089\u306A\u3044\u3053\u3068\u3067\u3042\u308B\u3002 \u6B63\u5247\u306A\u9806\u5E8F\u6570\u306F\u59CB\u9806\u5E8F\u6570 (en:initial ordinal) \u3067\u3042\u308B\u3002\u3057\u304B\u3057\u3001\u59CB\u9806\u5E8F\u6570\u3060\u304B\u3089\u3068\u3044\u3063\u3066\u6B63\u5247\u3067\u3042\u308B\u3068\u306F\u9650\u3089\u306A\u3044\u3002 \u6B63\u5247\u3067\u306A\u3044\u6574\u5217\u7121\u9650\u96C6\u5408\u306E\u6FC3\u5EA6\u306F\u7279\u7570\u57FA\u6570\u3068\u547C\u3070\u308C\u308B\u3002 \u6709\u9650\u9806\u5E8F\u6570\u306B\u5BFE\u3057\u3066\u306F\u666E\u901A\u3001\u6B63\u5247\u3084\u7279\u7570\u3068\u8A00\u3063\u305F\u547C\u3073\u65B9\u306F\u3057\u306A\u3044\u3002"@ja . "Regular cardinal"@en . . . . . . . "Regularna liczba kardynalna \u2013 niesko\u0144czona liczba kardynalna, kt\u00F3ra nie mo\u017Ce by\u0107 przedstawiona jako suma mniej ni\u017C \u03BA zbior\u00F3w mocy mniejszej ni\u017C \u03BA. Niesko\u0144czone liczby kardynalne kt\u00F3re nie s\u0105 regularne nazywamy liczbami singularnymi. W dalszej cz\u0119\u015Bci tego artyku\u0142u zak\u0142adamy ZFC. (Bez AC, niekt\u00F3re z definicji nale\u017Cy sformu\u0142owa\u0107 inaczej i niekt\u00F3re stwierdzenia nie s\u0105 prawdziwe)."@pl . . . "In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular. In the presence of the axiom of choice, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal : 1. \n* is a regular cardinal. 2. \n* If and for all , then . 3. \n* If , and if and for all , then . 4. \n* The category of sets of cardinality less than and all functions between them is closed under colimits of cardinality less than . 5. \n* is a regular ordinal (see below) Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts. The situation is slightly more complicated in contexts where the axiom of choice might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only. An infinite ordinal is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type less than . A regular ordinal is always an initial ordinal, though some initial ordinals are not regular, e.g., (see the example below)."@en . . "En teoria de conjunts, un cardinal regular \u00E9s un nombre cardinal que \u00E9s igual a la seva pr\u00F2pia cofinalitat. M\u00E9s expl\u00EDcitament, aix\u00F2 significa que \u00E9s un cardinal normal si i nom\u00E9s si cada subconjunt il\u00B7limitat t\u00E9 cardinalitat . Els infinits cardinals ben ordenats que no s\u00F3n regulars s\u2019anomenen cardinals singulars. Els nombres cardinals finits normalment no s\u2019anomenen ni regulars ni singulars. En pres\u00E8ncia de l'axioma d'elecci\u00F3, qualsevol nombre cardinal pot estar ben ordenat i, i llavors els seg\u00FCents s\u00F3n equivalents per a un cardinal :"@ca . . "Cardinal regular"@ca . . "Regulier kardinaalgetal"@nl . . . . . . "Regul\u00E1rn\u00ED ordin\u00E1l"@cs . . . . "Em matem\u00E1tica, especialmente em teoria de conjuntos, um cardinal \u00E9 denominado regular se ele \u00E9 igual a sua pr\u00F3pria cofinalidade. Caso contr\u00E1rio, \u00E9 dito singular."@pt . . . . . . "\u96C6\u5408\u8AD6\u306B\u304A\u3044\u3066\u3001\u6B63\u5247\u57FA\u6570\uFF08\u305B\u3044\u305D\u304F\u304D\u3059\u3046\u3001\u82F1: regular cardinal\uFF09\u3068\u306F\u3001\u305D\u306E\u5171\u7D42\u6570\u304C\u305D\u308C\u81EA\u8EAB\u3067\u3042\u308B\u57FA\u6570\u306E\u3053\u3068\u3002 \u7C21\u5358\u306B\u8A00\u3048\u3070\u3001\u6B63\u5247\u57FA\u6570\u306F\u5C0F\u3055\u3044\u30D1\u30FC\u30C4\u306E\u5C11\u306A\u3044\u96C6\u307E\u308A\u306B\u5206\u5272\u3067\u304D\u306A\u3044\u3082\u306E\u3067\u3042\u308B\u3002 (\u3053\u306E\u72B6\u6CC1\u306F\u9078\u629E\u516C\u7406\u3092\u4EEE\u5B9A\u3057\u306A\u3044\u6587\u8108\u3067\u306F\u3082\u3063\u3068\u8907\u96D1\u3067\u3042\u308B\u3002\u305D\u306E\u3088\u3046\u306A\u5834\u5408\u3001\u5168\u3066\u306E\u6FC3\u5EA6\u304C\u6574\u5217\u96C6\u5408\u306E\u6FC3\u5EA6\u3068\u306F\u9650\u3089\u306A\u304F\u3001\u4E0A\u8A18\u306E\u5B9A\u7FA9\u306F\u6574\u5217\u96C6\u5408\u306E\u6FC3\u5EA6\u306E\u307F\u306B\u5BFE\u3057\u3066\u306A\u3055\u308C\u308B\u3002) \u9078\u629E\u516C\u7406\u3092\u4EEE\u5B9A\u3059\u308B\u3068\u304D\u306F\u3001\u3044\u304B\u306A\u308B\u6FC3\u5EA6\u3082\u57FA\u6570\u306B\u306A\u308A\u3001\u7121\u9650\u57FA\u6570 \u304C\u6B63\u5247\u3067\u3042\u308B\u3053\u3068\u306F \u672A\u6E80\u306E\u57FA\u6570\u306E \u672A\u6E80\u500B\u306E\u548C\u3067\u306F\u8868\u305B\u306A\u3044\u3053\u3068\u3068\u540C\u5024\u306B\u306A\u308B\u3002 \u307E\u305F\u3001\u7121\u9650\u9806\u5E8F\u6570 \u304C\u6B63\u5247\u9806\u5E8F\u6570\u3068\u547C\u3070\u308C\u308B\u306E\u306F\u3001\u305D\u308C\u304C\u6975\u9650\u9806\u5E8F\u6570\u3067\u3088\u308A\u5C0F\u3055\u3044\u9806\u5E8F\u6570\u306E\u9806\u5E8F\u578B\u304C \u672A\u6E80\u306E\u96C6\u5408\u306E\u6975\u9650\u306B\u306A\u3089\u306A\u3044\u3053\u3068\u3067\u3042\u308B\u3002 \u6B63\u5247\u306A\u9806\u5E8F\u6570\u306F\u59CB\u9806\u5E8F\u6570 (en:initial ordinal) \u3067\u3042\u308B\u3002\u3057\u304B\u3057\u3001\u59CB\u9806\u5E8F\u6570\u3060\u304B\u3089\u3068\u3044\u3063\u3066\u6B63\u5247\u3067\u3042\u308B\u3068\u306F\u9650\u3089\u306A\u3044\u3002 \u6B63\u5247\u3067\u306A\u3044\u6574\u5217\u7121\u9650\u96C6\u5408\u306E\u6FC3\u5EA6\u306F\u7279\u7570\u57FA\u6570\u3068\u547C\u3070\u308C\u308B\u3002 \u6709\u9650\u9806\u5E8F\u6570\u306B\u5BFE\u3057\u3066\u306F\u666E\u901A\u3001\u6B63\u5247\u3084\u7279\u7570\u3068\u8A00\u3063\u305F\u547C\u3073\u65B9\u306F\u3057\u306A\u3044\u3002"@ja . . . . . "En teor\u00EDa de conjuntos, un ordinal regular es un ordinal que satisface una propiedad especial de \"clausura\", a saber, que s\u00F3lo puede ser aproximado por un conjunto de ordinales m\u00E1s peque\u00F1os que \u00E9l, si este conjunto tiene un cardinal inferior al propio ordinal que se pretende aproximar. Los ordinales que satisfacen esa condici\u00F3n son ordinales regulares, formalmente son la clase formada por: En particular todos los ordinales regulares son cardinales. Un resultado bien conocido de la teor\u00EDa de conjuntos es que la clase de todos los cardinales al igual que la clase de todos los ordinales regulares son clases no acotadas contenidas en los ordinales. Para cualquier ordinal \u03B1 existe un cardinal m\u00EDnimo \u03B1+ que es mayor que \u03B1. Todos los cardinales de la forma \u03B1+ son ordinales regulares. La noci\u00F3n de acotaci\u00F3n aqu\u00ED es la siguiente: \n* Datos: Q1193137"@es . . . . "374128"^^ . "En teor\u00EDa de conjuntos, un ordinal regular es un ordinal que satisface una propiedad especial de \"clausura\", a saber, que s\u00F3lo puede ser aproximado por un conjunto de ordinales m\u00E1s peque\u00F1os que \u00E9l, si este conjunto tiene un cardinal inferior al propio ordinal que se pretende aproximar. Los ordinales que satisfacen esa condici\u00F3n son ordinales regulares, formalmente son la clase formada por: \n* Datos: Q1193137"@es . . "Cardinal regular"@es . . . "8310"^^ . . "Regularna liczba kardynalna \u2013 niesko\u0144czona liczba kardynalna, kt\u00F3ra nie mo\u017Ce by\u0107 przedstawiona jako suma mniej ni\u017C \u03BA zbior\u00F3w mocy mniejszej ni\u017C \u03BA. Niesko\u0144czone liczby kardynalne kt\u00F3re nie s\u0105 regularne nazywamy liczbami singularnymi. W dalszej cz\u0119\u015Bci tego artyku\u0142u zak\u0142adamy ZFC. (Bez AC, niekt\u00F3re z definicji nale\u017Cy sformu\u0142owa\u0107 inaczej i niekt\u00F3re stwierdzenia nie s\u0105 prawdziwe)."@pl . . . "In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular. In the presence of the axiom of choice, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal :"@en . . "Cardinal r\u00E9gulier"@fr . . . . . "En teoria de conjunts, un cardinal regular \u00E9s un nombre cardinal que \u00E9s igual a la seva pr\u00F2pia cofinalitat. M\u00E9s expl\u00EDcitament, aix\u00F2 significa que \u00E9s un cardinal normal si i nom\u00E9s si cada subconjunt il\u00B7limitat t\u00E9 cardinalitat . Els infinits cardinals ben ordenats que no s\u00F3n regulars s\u2019anomenen cardinals singulars. Els nombres cardinals finits normalment no s\u2019anomenen ni regulars ni singulars. En pres\u00E8ncia de l'axioma d'elecci\u00F3, qualsevol nombre cardinal pot estar ben ordenat i, i llavors els seg\u00FCents s\u00F3n equivalents per a un cardinal : 1. \n* \u00E9s un cardinal regular. 2. \n* Si i per tot , llavors . 3. \n* Si , i si i per tot , llavors . 4. \n* La categoria de conjunts de cardinalitat inferior que i totes les funcions entre ells s\u00F3n tancades sota colimits de cardinalitat menor que . B\u00E0sicament, aix\u00F2 significa que un cardinal regular \u00E9s aquell que no es pot desglossar en un petit nombre de parts m\u00E9s petites. La situaci\u00F3 \u00E9s lleugerament m\u00E9s complicada en contextos on l'axioma d'elecci\u00F3 podria fallar, mentre en aquest cas no tots els cardinals s\u00F3n necess\u00E0riament les cardinalitats de conjunts ben ordenats. En aquest cas, l'equival\u00E8ncia anterior nom\u00E9s es pot aplicar als cardinals ben ordenats. Un ordinal infinit \u00E9s un ordinal regular si \u00E9s un ordinal de l\u00EDmit que no \u00E9s el l\u00EDmit d'un conjunt d'ordinals m\u00E9s petits que com al conjunt t\u00E9 menys de . Un ordinal regular \u00E9s sempre un ordinal inicial, encara que alguns no s\u00F3n regulars, p. ex., (veu l'exemple a sota). Un ordinal infinit \u00E9s un ordinal regular si es tracta d\u2019un ordinal l\u00EDmit que no \u00E9s el l\u00EDmit d\u2019un conjunt d\u2019ordinals m\u00E9s petits que, com a conjunt, tenen un tipus d\u2019ordre inferior a . Un ordinal regular sempre \u00E9s un ordinal inicial, tot i que alguns ordinals inicials no s\u00F3n regulars, per exemple, (vegeu l'exemple seg\u00FCent)."@ca . . . "Cardinais regulares e singulares"@pt . "Regularna liczba kardynalna"@pl . . . "\u6B63\u5247\u57FA\u6570"@ja . . . "Regul\u00E1rn\u00ED ordin\u00E1l (tak\u00E9 regul\u00E1rn\u00ED kardin\u00E1l) je matematick\u00FD pojem z oblasti teorie mno\u017Ein (ordin\u00E1ln\u00ED aritmetiky)."@cs . "In de verzamelingenleer, een deelgebied van de wiskunde, is een regulier kardinaalgetal een kardinaalgetal, dat gelijk is aan haar eigen cofinaliteit. Dus informeel gesproken is een regulier kardinaalgetal er eentje dat niet kan worden opgesplitst in een kleinere collectie van kleinere delen."@nl . . "Regul\u00E1rn\u00ED ordin\u00E1l (tak\u00E9 regul\u00E1rn\u00ED kardin\u00E1l) je matematick\u00FD pojem z oblasti teorie mno\u017Ein (ordin\u00E1ln\u00ED aritmetiky)."@cs . "In de verzamelingenleer, een deelgebied van de wiskunde, is een regulier kardinaalgetal een kardinaalgetal, dat gelijk is aan haar eigen cofinaliteit. Dus informeel gesproken is een regulier kardinaalgetal er eentje dat niet kan worden opgesplitst in een kleinere collectie van kleinere delen."@nl . . . . . . "\uC815\uCE59 \uAE30\uC218"@ko .