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ce to use singular and remove ambiguity; link antisymmetric and asymmetric; these claims need to be sourced
 
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In [[set theory]], a '''regular cardinal''' is a  [[cardinal number]] that is equal to its own [[cofinality]]. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts.
== 強い爪魔法皇帝の手まで、虚数をクリックします ==


If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal <math>\kappa</math> is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than <math>\kappa</math>, the elements of which are cardinals less than <math>\kappa</math>. (The situation is slightly more complicated in contexts where the [[axiom of choice]] might fail; in that case not all cardinals are necessarily the cardinalities of [[well-ordered]] sets. In that case, the above definition is restricted to well-orderable cardinals only.)
やれやれ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン 偽物]！<br>ローブは、強力な爪を<br>魔法皇帝の炎の水門が離れて、​​突然怖い無数の見物人が急速エスケープと呪わすべての方向に注ぐ洪水の高騰かのように、ロールアップ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 価格]。<br><br>強い爪魔法皇帝の手まで、虚数をクリックします [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 店舗]。<br><br>突然......<br>モーメントの<br>数十スコープ周囲の空間がまだ立っているようだ、など魔法皇帝など皇帝西安周の言語、平行異なる個人的な空間が同じのまわりに拘留されているようだと感じ、速度は状態の非常に敵に低下します。<br><br>「ダークのマスターはその後、私はあなたを強制しようかではありません。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン 銀座] 'マジック天皇強力な爪は手が突然現れ爪は、皮膚表面をにじみ出る赤い赤い光が、すぐに指をカバー......<br><br>二十から八章の危機<br>マジック皇帝が強い爪魔法皇帝であるとして<br>強い爪が知られている、それは非常に鋭い爪フランスの攻撃の彼の神秘的なセットがあった。<br>「ダイに行く [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン スニーカー]。「強い爪魔法皇帝のチャタリング奇妙な笑顔、彼の手のプローブクロー型にそれを殺すために最も弱いデュオミ皇帝西安に向かって<br>が、周大福罪の言語はすぐに駆けつけかわす、
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</ul>


An infinite ordinal <math>\alpha</math> is regular [[if and only if]] it is a [[limit ordinal]] which is not the limit of a set of smaller ordinals which set has [[order type]] less than <math>\alpha</math>. A regular ordinal is always an [[initial ordinal]], though some initial ordinals are not regular.
== 二年未満の短い期間を ==


Infinite well-ordered cardinals which are not regular are called '''singular cardinals'''. Finite cardinal numbers are typically not called regular or singular.
単にアリです。霊の世界がアリよりもはるかに少ないですがと。<br><br>しかし週は本当に期待していなかった [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン 銀座]。<br>二年未満の短い期間を<br> [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン サイズ]。<br><br>二年間、神のために、業界の専門家は、それは彼がしたこの短い時間で非常に短い時間であることが一つの大きなショットを成長させるために定命の世界はほとんどの人も、霊の世界の被害にもかかわらず [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン バッグ]。<br><br>ドンヒョンジンShanlan非常に神秘的な寺院Dianzhu、精錬マスタの新学期。<br>鉱山の状況について<br>シングルは間違いなく彼の美しい町プリンス低いよりも優れています。<br><br>と人気で、精錬マスターは、間違いなくその党軍か否かの主要国に歓迎され、敵対的な精錬マスターではありません。<br><br>'これはああにしても2年ではありません [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ]！'週は大きな心が嫌いだった。<br>武道、少し定命の世界で目の点滅を<br>、霊の世界は精錬マスターに崇め状態になっています [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタンジャパン]！<br><br>「秦ゆう、これは私のいとこと競合する多大な努力を尽くさなければ確かにあるが、ただ、いとこは最終的に必ず私に属し、どんなに
__NOTOC__
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== Examples ==
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</ul>


The ordinals less than <math>\omega</math> are finite. A finite sequence of finite ordinals always has a finite maximum, so <math>\omega</math> cannot be the limit of any sequence of type less than <math>\omega</math> whose elements are ordinals less than <math>\omega</math>, and is therefore a regular ordinal. <math>\aleph_0</math> ([[aleph-null]]) is a regular cardinal because its initial ordinal, <math>\omega</math>, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.
== Sheng Huang Dong extremely long exhortations over ==


<math>\omega+1</math> is the [[successor ordinal|next ordinal number]] greater than <math>\omega</math>. It is singular, since it is not a limit ordinal. <math>\omega+\omega</math> is the next limit ordinal after <math>\omega</math>. It can be written as the limit of the sequence <math>\omega</math>, <math>\omega+1</math>, <math>\omega+2</math>, <math>\omega+3</math>, and so on. This sequence has order type <math>\omega</math>, so <math>\omega+\omega</math> is the limit of a sequence of type less than <math>\omega+\omega</math> whose elements are ordinals less than <math>\omega+\omega</math>, therefore it is singular.
Which the refining fire temple, always in [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン メンズ 靴] time to accelerate to limit state, the time constant acceleration. Qin Yu estimated [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン 値段] that they should test the Wuliubaiwan years.<br><br>'material enough? how to do it?' Qin Yu frown thought.<br><br>suddenly Qin Yu laughed: 'I'd forgotten, Sheng Huang Dong extremely long exhortations over, the material is [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン 銀座] not enough to want him ah!' Qin Yuli mind acoustic to re-let in autumn autumn Chung Chung Fu personally. to receive a large number of ordinary materials.<br><br>'It seems to go back to the house of a mysterious trip to Lan.' Qin Yu said to himself.<br><br>'Huh?' Qin Yu suddenly hesitated, 'I reached the upper part of the realm of the gods?'<br><br>millions of years ago. Qin Yu completely immersed in the mixing device, but also did not notice that they have reached the upper part of the realm of the gods, which millions of years ago, Qin Yu 'Hualianhang avatar' has left the space in the left eyebrow fall sentiment.<br><br>down millions of years, Qin Yu [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン 日本] had left half the space to spare sentiment, Hualianhang avatar, has the sentiment of the half.<br><br>Qin Yu to spare [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン バッグ] in that mid-air
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<math>\aleph_1</math> is the [[successor cardinal|next cardinal number]] greater than <math>\aleph_0</math>, so the cardinals less than <math>\aleph_1</math> are [[countable set|countable]] (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So <math>\aleph_1</math> cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
== is still observed in whole pieces Dark Star region. ==


<math>\aleph_\omega</math> is the next cardinal number after the sequence <math>\aleph_0</math>, <math>\aleph_1</math>, <math>\aleph_2</math>, <math>\aleph_3</math>, and so on. Its initial ordinal <math>\omega_\omega</math> is the limit of the sequence <math>\omega</math>, <math>\omega_1</math>, <math>\omega_2</math>, <math>\omega_3</math>, and so on, which has order type <math>\omega</math>, so <math>\omega_\omega</math> is singular, and so is <math>\aleph_\omega</math>. Assuming the axiom of choice, <math>\aleph_\omega</math> is the first infinite cardinal which is singular (the first infinite ''ordinal'' which is singular is <math>\omega+1</math>). Proving the existence of singular cardinals requires the [[axiom schema of replacement|axiom of replacement]], and in fact the inability to prove the existence of <math>\aleph_\omega</math> in [[Zermelo set theory]] is what led [[Adolf Abraham Halevi Fraenkel|Fraenkel]] to postulate this axiom.{{Citation needed|date=August 2011}}
. Should be considered in accordance with the gravity extremely [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京] scary ...... Qin Yu estimates, should catch up with the mortal world of 'at Star' was. But myself as a person, but they can live to go on top.<br><br>cells [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン メンズ 靴] completely collapse, seems to have been, but after cell transformation, now more energy cells. Fully able [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 店舗] to withstand the foot of the dark star's gravity.<br><br>Qin Yu diffuse the power of the soul open to.<br><br>'Well? This Dark Star was able to attract the power of the soul?' Qin Yu [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン 値段] was shocked to find that even the dark star of horror gravitational influence of the power of the soul, but a meteor tears assisted Qin Yu, is still observed in whole pieces Dark Star region.<br><br>dark star is very small.<br><br>accurate to say, dark stars are generally small, dark star also about a thousand miles in diameter. If you are an ordinary mortal planet, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン サンダル] Qin Yu passed the estimated blink of an eye.<br><br>but now? with walking speed, walking a thousand miles. How long? One day if Barry, I guess we need a full ten days now. And that is only the diameter of it, around the dark stars in a circle, estimate
 
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== Properties ==
<ul>
 
 
Uncountable [[limit cardinal]]s that are also regular are known as weakly [[inaccessible cardinals]]. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily [[fixed point (mathematics)|fixed point]]s of the [[aleph number|aleph function]], though not all fixed points are regular.  For instance, the first fixed point is the limit of the <math>\omega</math>-sequence <math>\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}}, ...</math> and is therefore singular.
  <li>[http://wedding.stanandlisa.com/cgi-bin/guestbook/guestbook.cgi http://wedding.stanandlisa.com/cgi-bin/guestbook/guestbook.cgi]</li>
 
 
If the [[axiom of choice]] holds, then every [[successor cardinal]] is regular.  Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinal numbers cannot be proven to be equal to any particular aleph, for instance the [[cardinality of the continuum]], whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see [[Easton's theorem]]). The [[continuum hypothesis]] postulates that the cardinality of the continuum is equal to <math>\aleph_1</math> which is regular.
  <li>[http://zhongguodingzhi.com/forum.php?mod=viewthread&tid=653969 http://zhongguodingzhi.com/forum.php?mod=viewthread&tid=653969]</li>
 
 
Without the axiom of choice, there would be cardinal numbers which were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore only the [[aleph number]]s can meaningfully be called regular or singular cardinals. Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need not be countable. It is consistent with [[Zermelo–Fraenkel set theory|ZF]] that <math>\omega_1</math> be the limit of a countable sequence of countable ordinals as well as the set of real numbers is countable union of countable sets. Furthermore, it is consistent with ZF that every aleph bigger than <math>\aleph_0</math> is singular (a result proved by [[Moti Gitik]]).
  <li>[http://bbs.jyhyly.com/forum.php?mod=viewthread&tid=87922 http://bbs.jyhyly.com/forum.php?mod=viewthread&tid=87922]</li>
 
 
== See also ==
</ul>
* [[Inaccessible cardinal]]
 
== References ==
* {{aut|[[Herbert Enderton|Herbert B. Enderton]]}}, ''Elements of Set Theory'', ISBN 0-12-238440-7
* {{aut|[[Kenneth Kunen]]}}, ''Set Theory, An Introduction to Independence Proofs'', ISBN 0-444-85401-0
 
[[Category:Ordinal numbers]]
[[Category:Cardinal numbers]]

Latest revision as of 19:00, 3 August 2014

強い爪魔法皇帝の手まで、虚数をクリックします

やれやれ クリスチャンルブタン 偽物！
ローブは、強力な爪を
魔法皇帝の炎の水門が離れて、​​突然怖い無数の見物人が急速エスケープと呪わすべての方向に注ぐ洪水の高騰かのように、ロールアップ クリスチャンルブタン 価格。

強い爪魔法皇帝の手まで、虚数をクリックします クリスチャンルブタン 店舗。

突然......
モーメントの
数十スコープ周囲の空間がまだ立っているようだ、など魔法皇帝など皇帝西安周の言語、平行異なる個人的な空間が同じのまわりに拘留されているようだと感じ、速度は状態の非常に敵に低下します。

「ダークのマスターはその後、私はあなたを強制しようかではありません。 クリスチャンルブタン 銀座 'マジック天皇強力な爪は手が突然現れ爪は、皮膚表面をにじみ出る赤い赤い光が、すぐに指をカバー......

二十から八章の危機
マジック皇帝が強い爪魔法皇帝であるとして
強い爪が知られている、それは非常に鋭い爪フランスの攻撃の彼の神秘的なセットがあった。
「ダイに行く クリスチャンルブタン スニーカー。「強い爪魔法皇帝のチャタリング奇妙な笑顔、彼の手のプローブクロー型にそれを殺すために最も弱いデュオミ皇帝西安に向かって
が、周大福罪の言語はすぐに駆けつけかわす、 相关的主题文章：

二年未満の短い期間を

単にアリです。霊の世界がアリよりもはるかに少ないですがと。

しかし週は本当に期待していなかった クリスチャンルブタン 銀座。
二年未満の短い期間を
クリスチャンルブタン サイズ。

二年間、神のために、業界の専門家は、それは彼がしたこの短い時間で非常に短い時間であることが一つの大きなショットを成長させるために定命の世界はほとんどの人も、霊の世界の被害にもかかわらず クリスチャンルブタン バッグ。

ドンヒョンジンShanlan非常に神秘的な寺院Dianzhu、精錬マスタの新学期。
鉱山の状況について
シングルは間違いなく彼の美しい町プリンス低いよりも優れています。

と人気で、精錬マスターは、間違いなくその党軍か否かの主要国に歓迎され、敵対的な精錬マスターではありません。

'これはああにしても2年ではありません クリスチャンルブタン ブーツ！'週は大きな心が嫌いだった。
武道、少し定命の世界で目の点滅を
、霊の世界は精錬マスターに崇め状態になっています クリスチャンルブタンジャパン！

「秦ゆう、これは私のいとこと競合する多大な努力を尽くさなければ確かにあるが、ただ、いとこは最終的に必ず私に属し、どんなに 相关的主题文章：

Sheng Huang Dong extremely long exhortations over

Which the refining fire temple, always in クリスチャンルブタン メンズ 靴 time to accelerate to limit state, the time constant acceleration. Qin Yu estimated クリスチャンルブタン 値段 that they should test the Wuliubaiwan years.

'material enough? how to do it?' Qin Yu frown thought.

suddenly Qin Yu laughed: 'I'd forgotten, Sheng Huang Dong extremely long exhortations over, the material is クリスチャンルブタン 銀座 not enough to want him ah!' Qin Yuli mind acoustic to re-let in autumn autumn Chung Chung Fu personally. to receive a large number of ordinary materials.

'It seems to go back to the house of a mysterious trip to Lan.' Qin Yu said to himself.

'Huh?' Qin Yu suddenly hesitated, 'I reached the upper part of the realm of the gods?'

millions of years ago. Qin Yu completely immersed in the mixing device, but also did not notice that they have reached the upper part of the realm of the gods, which millions of years ago, Qin Yu 'Hualianhang avatar' has left the space in the left eyebrow fall sentiment.

down millions of years, Qin Yu クリスチャンルブタン 日本 had left half the space to spare sentiment, Hualianhang avatar, has the sentiment of the half.

Qin Yu to spare クリスチャンルブタン バッグ in that mid-air 相关的主题文章：

is still observed in whole pieces Dark Star region.

. Should be considered in accordance with the gravity extremely クリスチャンルブタン 店舗 東京 scary ...... Qin Yu estimates, should catch up with the mortal world of 'at Star' was. But myself as a person, but they can live to go on top.

cells クリスチャンルブタン メンズ 靴 completely collapse, seems to have been, but after cell transformation, now more energy cells. Fully able クリスチャンルブタン 店舗 to withstand the foot of the dark star's gravity.

Qin Yu diffuse the power of the soul open to.

'Well? This Dark Star was able to attract the power of the soul?' Qin Yu クリスチャンルブタン 値段 was shocked to find that even the dark star of horror gravitational influence of the power of the soul, but a meteor tears assisted Qin Yu, is still observed in whole pieces Dark Star region.

dark star is very small.

accurate to say, dark stars are generally small, dark star also about a thousand miles in diameter. If you are an ordinary mortal planet, クリスチャンルブタン サンダル Qin Yu passed the estimated blink of an eye.

but now? with walking speed, walking a thousand miles. How long? One day if Barry, I guess we need a full ten days now. And that is only the diameter of it, around the dark stars in a circle, estimate 相关的主题文章：

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