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| In [[mathematical logic]], the '''Löwenheim–Skolem theorem''', named for [[Leopold Löwenheim]] and [[Thoralf Skolem]], states that if a countable first-order [[Theory (mathematical logic)|theory]] has an infinite [[Interpretation (logic)|model]], then for every infinite [[cardinal number]] κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model [[up to isomorphism]].
| | == to be able to eventually decay I think hundreds of years == |
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| The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the [[compactness theorem]], that are used in [[Lindström's theorem]] to characterize [[first-order logic]]. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as [[second-order logic]].
| | Silent children, elders slowly opening day sentence: 'A treasure is to be attained become Dao, longevity Fam trivial matter experts to refine Tao, nor is it an easy thing Fortunately, your two pieces of magic. device,[http://www.aseanacity.com/webalizer/prada-bags-20.html プラダ 財布 迷彩], the foundation has a very deep, but five prison Ong Teng function is too complex to excel Dao,[http://www.aseanacity.com/webalizer/prada-bags-32.html プラダ 迷彩 財布], also must first gather eight million demons, then by hundreds of years of longevity Fam master fine blood exercise, to be able to eventually decay I think hundreds of years,[http://www.aseanacity.com/webalizer/prada-bags-29.html プラダ スタッズ 財布], and so you may not be right. '<br>'He was right, five prison to become Tao Wang Ding,[http://www.aseanacity.com/webalizer/prada-bags-35.html プラダ 財布 レディース], is not an easy thing.' Yan Fang cold in the body,[http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ最新財布], issued a voice.<br>'hundreds of years, really can not wait. Modao blood firmament that it?' Fang Han thought, is a fact.<br>'top quality Modao blood firmament is still five prison Wang Ding, only a single function to pure destruction known.'d better refining some, but I still want to spend a hundred years to |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.olpcnews.com/mt/mt-search.cgi and everyone saw that Fengsha]</li> |
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| | <li>[http://oldolavianslodge.org.uk/cgi-bin/old-olavians-lodge/active_guestbook/guestbook.cgi ' プリンスシニオラの]</li> |
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| | <li>[http://www.tldxzl.com/plus/feedback.php?aid=43 ' 多くの真理の古代帝国の神社]</li> |
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| | </ul> |
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| == Background == | | == seems to be well-intentioned == |
| A [[signature (logic)|signature]] consists of a set of function symbols ''S''<sub>func</sub>, a set of relation symbols ''S''<sub>rel</sub>, and a function <math>\operatorname{ar}: S_{\operatorname{func}}\cup S_{\operatorname{rel}} \rightarrow \mathbb{N}_0</math> representing the [[arity]] of function and relation symbols. (A nullary function symbol is called a constant symbol.) In the context of first-order logic, a signature is sometimes called a '''language'''. It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains.
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| A first-order '''[[Theory (mathematical logic)|theory]]''' consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature. Theories are often specified by giving a list of axioms that generate the theory, or by giving a structure and taking the theory to consist of the sentences satisfied by the structure.
| | Green,[http://www.aseanacity.com/webalizer/prada-bags-23.html prada 財布 通販], one of a witch word around his body suspended.<br>'big secret witch curse.'<br>side cold saw this old book that demon beneath books,[http://www.aseanacity.com/webalizer/prada-bags-29.html prada 財布 スタッズ], old witch word emerged out of the book is a Mantra. Belong to something 巫道文 Ming.<br>'young, make haste to leave here not your place to belong elusive dòng days of restricted areas. after that there exists a wide variety of powerful, even feathering mén saints, but also not in depth which from the beginning to now, only 华天君 one day be able to smooth the elusive dòng being I was a kindness. '<br>that body withered,[http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ], like the high priest of the old man saw the party generally cold, his eyes suddenly transform into kind, seems to be well-intentioned, good persuade parties to go cold.<br>'kindness,[http://www.aseanacity.com/webalizer/prada-bags-35.html プラダ スタッズ 財布]? give me out!'<br>side cold but does not see this old man, suddenly turned to look behind him.<br>Jie Jie,[http://www.aseanacity.com/webalizer/prada-bags-26.html 財布 プラダ レディース], Jie Jie |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://souk.tidjelabine.net/index.php?page=item&id=410814 徹底的悪魔混乱、混乱している]</li> |
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| | <li>[http://www.cccpress.com/big5/?action-viewcomment-type-news-itemid-1988 一度理解]</li> |
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| | <li>[http://www.yy0536.com/plus/feedback.php?aid=543 側面冷たい顔]</li> |
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| | </ul> |
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| Given a signature σ, a σ-[[structure (mathematical logic)|structure]] ''M''
| | == その中に私と一緒に行く == |
| is a concrete interpretation of the symbols in σ. It consists of an underlying set (often also denoted by "''M''") together with an interpretation of the function and relation symbols of σ. An interpretation of a constant symbol of σ in ''M'' is simply an element of ''M''. More generally, an interpretation of an ''n''-ary function symbol ''f'' is a function from ''M''<sup>''n''</sup> to ''M''. Similarly, an interpretation of a relation symbol ''R'' is an ''n''-ary relation on ''M'', i.e. a subset of ''M''<sup>''n''</sup>.
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| A '''substructure''' of a σ-structure ''M'' is obtained by taking a subset ''N'' of ''M'' which is closed under the interpretations of all the function symbols in σ (hence includes the interpretations of all constant symbols in σ), and then restricting the interpretations of the relation symbols to ''N''. An [[elementary substructure]] is a very special case of this; in particular an elementary substructure satisfies exactly the same first-order sentences as the original structure (its '''elementary extension''').
| | 市場のほとんどの縦断,[http://www.aseanacity.com/webalizer/prada-bags-22.html prada 新作 財布]。今、彼が言うすべてがそうしないと、理解の世界の外になり、私たちの小さな団体ではなく、定期的に礼拝に行く。 ' 「私は何かを購入したい！その中に私と一緒に行く,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダ レディース 財布]。 '<br><br>側風邪は僧侶が道の前で踊って、手を振った,[http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト]。 目の<br>きらめきの間に、それは素晴らしい平野、新しい建物の要塞の裁判所の前の海の水平線の上に上陸した。この都市は、高尚なまでの紺のドレスを着て、生意気1僧侶立って、町の門で、山のように立っている。<br>「ストップ,[http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ最新財布]！地平線シークラブの強豪を上陸させ、僧侶たちは歩くことになります。 '<br>低温側と光華、上空を飛ぶ重不滅フル10以上の長寿のマスターは、ドリンクを押し込んだ僧侶全体にこのクリーチャーを見ました。低温側と僧侶の道を生きて停止します,[http://www.aseanacity.com/webalizer/prada-bags-29.html プラダ新作バッグ2014]。 前任者 '<br>、怒らないで、古代の海の地平線地平線裁判所の命令は、統一された理解の世界を送信する |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.iyengar-yoga.com/perl/directory/user.cgi 、不死の間でUDLグローに入った]</li> |
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| | <li>[http://cgi.www5a.biglobe.ne.jp/~inline/cgi-bin/yybbs.cgi さらに、いくつかの男性と女性があり、すべての10を見て]</li> |
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| | <li>[http://www.wikimusicmemorabilia.com/User:Xqvmijqdpz#.C2.A0.E5.BD.BC.E3.82.89.E3.81.AF.E5.AE.8C.E5.85.A8.E3.81.AB.E6.9A.B4.E5.8A.9B.E7.9A.84.E3.80.81Dao.E3.81.AB.E6.8A.BC.E3.81.95.E3.82.8C.E3.80.81.E3.81.A9.E3.81.93.E3.81.A7.E7.9F.A5.E3.81.A3 彼らは完全に暴力的、Daoに押され、どこで知っ]</li> |
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| | </ul> |
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| == Precise statement == | | == オーダーの下 == |
| The modern statement of the theorem is both more general and stronger than the version for countable signatures stated in the introduction.
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| In its general form, the '''Löwenheim–Skolem Theorem''' states that for every [[signature (logic)|signature]] σ, every infinite σ-[[structure (mathematical logic)|structure]] ''M'' and every infinite cardinal number κ ≥ |σ|, there is a σ-structure ''N'' such that |''N''| = κ and
| | 大規模な配列が競合の下に配置することができる,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダ レディース 財布]。 オーダーの下<br>Gutianquanメインドア、立ち上がる。<br>バン,[http://www.aseanacity.com/webalizer/prada-bags-23.html prada スタッズ 財布]！ ストリップ<br>、空隙に登場し、突然武器を飛び出す彼の手のフリックは、古代の名残マップ、大気上記道路、雲に凝縮される,[http://www.aseanacity.com/webalizer/prada-bags-21.html プラダ 財布 スタッズ]。大気の無数の先祖の神々は、上記の回転、息の「祖先」で、パッシングは低温側の心に達した,[http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ]。<br>牙冷たい驚き 'これは.......です'： '！。トップグレードセント' 製品中の<br>セント、あなたは精錬することができるように、大羅ジン罪の法律を必要とし、トップグレードのセントは、あなたが成功する精錬することできるようにするには、より強力な先祖のセントを必要としています,[http://www.aseanacity.com/webalizer/prada-bags-23.html prada 財布 リボン]。<br>側風邪は、トップグレードのセントを見たことがない。<br>さて、このトップグレードGutianquanメインドア雰囲気の法則Cnの「前駆」のうちセント祭り、祖先、天の空隙を引き裂くのは簡単、ライブ群衆の守護者。<br>「これは、 |
| * if κ < |''M''| then ''N'' is an elementary substructure of ''M'';
| | 相关的主题文章： |
| * if κ > |''M''| then ''N'' is an elementary extension of ''M''.
| | <ul> |
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| | <li>[http://www.bemikitties.com/guestbook/guestbook.cgi 長寿ファム一緒に、彼は苦しむ作ったもの、確かに強力な存在]</li> |
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| | <li>[http://www.iehdi.org/wiki/index.php?title=User:Nyzovzbh#.E7.AC.AC.E4.B8.83.E7.99.BE.E5.9B.9B.E5.8D.81.E4.BA.94 第七百四十五]</li> |
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| | <li>[http://www.nicepowers.com/plus/view.php?aid=73431 「女の子を精緻化、あるあなたは]</li> |
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| | </ul> |
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| The theorem is often divided into two parts corresponding to the two bullets above. The part of the theorem asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the '''downward Löwenheim–Skolem Theorem'''. The part of the theorem asserting that a structure has elementary extensions of all larger cardinalities is known as the '''upward Löwenheim–Skolem Theorem'''.
| | == ドアのために、光大のまぐさとして記述することができます == |
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| The statement given in the introduction follows immediately by taking ''M'' to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. For historical variants of the theorem, see the notes below.
| | ドアのために、光大のまぐさとして記述することができます。<br>「あなたはいつも優しいをお楽しみいただけますことを保証するためにケースに私に従うならば低温側、私は弟子としてあなたを受け入れる,[http://www.aseanacity.com/webalizer/prada-bags-21.html prada 新作 財布]！私はケースを絶たない、美しい雲が、それらのすべては、比類のない美しさだ,[http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ]！ '終わりなき皇后はShouyi趙を入れて、9を見てYinjiuヤン2悪魔だけでなく、悪魔の神ではなく、条件のうち。 羽ヤシの扉を密閉し教えるhuanhangrnは今、ビッグFour'dは自分たちの生活マップを取得したい。それらの間に、明らかに仮止めではなく、1心の、自分の利益のために、それぞれ。<br>'ああ,[http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト]？現在、私は自分たちの生活は比較的良い数字れる専用入れ、最終的に4があるのですか,[http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ レディース]？'牙がそのような状況を見て寒さ、助けることができなかったが、チャネル」ではなく4、互いにコンテストで、最も高強度をマナの方は私とどのように彼のドアすべてで避難上のいくつかのシニア姉妹見習い,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダ人気財布]？ ' 私たちは古いペーストいるとき<br>「ハッハッハハッハッハ！ 'ナインヤン悪魔突然の笑いは、低温側を指して、「この小さな男は、私たちはまだ本当に、あなたを実行する |
| | | 相关的主题文章： |
| == Examples and consequences ==
| | <ul> |
| Let '''N''' denote the natural numbers and '''R''' the reals. It follows from the theorem that the theory of ('''N''', +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of ('''R''', +, ×, 0, 1) (the theory of [[real closed field]]s) has a countable model. There are, of course, axiomatizations characterizing ('''N''', +, ×, 0, 1) and ('''R''', +, ×, 0, 1) up to isomorphism. The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first-order. For example, the completeness of a linear order, which is used to characterize the real numbers as a complete ordered field, is a non-first-order property.
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| | | <li>[http://www.llxd688.com/home.php?mod=space&uid=52607 ]</li> |
| A theory is called '''categorical''' if it has only one model, up to isomorphism. This term was introduced by {{harvtxt|Veblen|1904}}, and for some time thereafter mathematicians hoped they could put mathematics on a solid foundation by describing a categorical first-order theory of some version of set theory. The Löwenheim–Skolem theorem dealt a first blow to this hope, as it implies that a first-order theory which has an infinite model cannot be categorical. Later, in 1931, the hope was shattered completely by [[Gödel's incompleteness theorem]].
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| | | <li>[http://www.llxd688.com/home.php?mod=space&uid=52607 ]</li> |
| Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of [[true arithmetic]], which satisfy every first-order [[Peano axioms|induction axiom]] but have non-inductive subsets. Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. This counterintuitive situation came to be known as [[Skolem's paradox]]; it shows that the notion of countability is not [[absoluteness (mathematical logic)|absolute]].
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| | | <li>[http://www.sxdyyqkyx.com/?action-viewcomment-type-news-itemid-11 冒険、経験の一つは、広大な叙事詩と呼ばれる]</li> |
| == Proof sketch ==
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| | | </ul> |
| === Downward part ===
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| For each first-order <math>\sigma \,</math>-formula <math>\varphi(y,x_{1}, \ldots, x_{n}) \,,</math> the [[axiom of choice]] implies the existence of a function
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| :<math>f_{\varphi}: M^n\to M</math> | |
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| such that, for all <math>a_{1}, \ldots, a_{n} \in M</math>, either
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| :<math>M\models\varphi(f_{\varphi} (a_1, \dots, a_n), a_1, \dots, a_n)</math>
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| or
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| :<math>M\models\neg\exists y \varphi(y, a_1, \dots, a_n) \,.</math>
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| Applying the axiom of choice again we get a function from the first order formulas <math>\varphi</math> to such functions <math>f_{\varphi} \,.</math>
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| The family of functions <math>f_{\varphi}</math> gives rise to a [[preclosure operator]] <math>F \,</math> on the [[power set]] of <math>M \,</math>
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| :<math>F(A) = \{b \in M \mid b = f_{\varphi}(a_1, \dots, a_n); \, \varphi \in \sigma ; \, a_1, \dots, a_n \in A \} </math> | |
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| for <math>A \subseteq M \,.</math>
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| Iterating <math>F \,</math> countably many times results in a [[closure operator]] <math>F^{\omega} \,.</math> Taking an arbitrary subset <math>A \subseteq M</math> such that <math>\left\vert A \right\vert = \kappa</math>, and having defined <math>N = F^{\omega}(A) \,,</math> one can see that also <math>\left\vert N \right\vert = \kappa \,.</math> <math>N \,</math> is an elementary substructure of <math>M \,</math> by the [[Tarski–Vaught test]].
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| The trick used in this proof is essentially due to Skolem, who introduced function symbols for the [[Skolem function]]s <math>f_{\varphi}</math> into the language. One could also define the <math>f_{\varphi}</math> as [[partial function]]s such that <math>f_{\varphi}</math> is defined if and only if <math>M \models \exists y \varphi(y,a_1,\dots,a_n) \,.</math> The only important point is that <math>F \,</math> is a preclosure operator such that <math>F(A) \,</math> contains a solution for every formula with parameters in <math>A \,</math> which has a solution in <math>M \,</math> and that
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| :<math>\left\vert F(A) \right\vert \leq \left\vert A \right\vert + \left\vert \sigma \right\vert + \aleph_0 \,.</math>
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| === Upward part ===
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| First, one extends the signature by adding a new constant symbol for every element of ''M''. The complete theory of ''M'' for the extended signature σ' is called the ''elementary diagram'' of ''M''. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of ''M'' the sentences ''c'' ≠ ''c<nowiki>'</nowiki>'' for any two distinct new constant symbols ''c'' and ''c<nowiki>'</nowiki>''. Using the [[compactness theorem]], the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model ''N'' which has cardinality exactly κ. It contains an isomorphic copy of ''M'' as an elementary substructure.
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| == Historical notes ==
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| This account is based mainly on {{harvtxt|Dawson|1993}}. To understand the early history of model theory one must distinguish between ''syntactical consistency'' (no contradiction can be derived using the deduction rules for first-order logic) and ''satisfiability'' (there is a model). Somewhat surprisingly, even before the [[Gödel's completeness theorem|completeness theorem]] made the distinction unnecessary, the term ''consistent'' was used sometimes in one sense and sometimes in the other.
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| The first significant result in what later became [[model theory]] was ''Löwenheim's theorem'' in [[Leopold Löwenheim]]'s publication "Über Möglichkeiten im Relativkalkül" (1915):
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| :For every countable signature σ, every σ-sentence which is satisfiable is satisfiable in a countable model.
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| Löwenheim's paper was actually concerned with the more general Peirce–Schröder [[calculus of relatives]] ([[relation algebra]] with quantifiers). He also used the now antiquated notations of [[Ernst Schröder]]. For a summary of the paper in English and using modern notations see {{harvtxt|Brady|2000|loc=chapter 8}}.
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| According to the received historical view, Löwenheim's proof was faulty because it implicitly used [[König's lemma]] without proving it, although the lemma was not yet a published result at the time. In a [[Historical revisionism|revisionist]] account, {{harvtxt|Badesa|2004}} considers that Löwenheim's proof was complete.
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| {{harvtxt|Skolem|1920}} gave a (correct) proof using formulas in what would later be called ''Skolem normal form'' and relying on the axiom of choice:
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| :Every countable theory which is satisfiable in a model ''M'', is satisfiable in a countable substructure of ''M''.
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| {{harvtxt|Skolem|1923}} also proved the following weaker version without the axiom of choice:
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| : Every countable theory which is satisfiable in a model is also satisfiable in a countable model.
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| {{harvtxt|Skolem|1929}} simplified {{harvtxt|Skolem|1920}}. Finally, [[Anatoly Maltsev|Anatoly Ivanovich Maltsev]] (Анато́лий Ива́нович Ма́льцев, 1936) proved the Löwenheim–Skolem theorem in its full generality {{harv|Maltsev|1936}}. He cited a note by Skolem, according to which the theorem had been proved by [[Alfred Tarski]] in a seminar in 1928. Therefore the general theorem is sometimes known as the ''Löwenheim–Skolem–Tarski theorem''. But Tarski did not remember his proof, and it remains a mystery how he could do it without the [[compactness theorem]].
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| It is somewhat ironic that Skolem's name is connected with the upward direction of the theorem as well as with the downward direction:
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| :''"I follow custom in calling Corollary 6.1.4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable sets."'' – {{harvtxt|Hodges|1993}}.
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| :''"Skolem [...] rejected the result as meaningless; Tarski [...] very reasonably responded that Skolem's formalist viewpoint ought to reckon the downward Löwenheim-Skolem theorem meaningless just like the upward."'' – {{harvtxt|Hodges|1993}}.
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| :''"Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence."'' – {{harvtxt|Poizat|2000}}.
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| == References ==
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| The Löwenheim–Skolem theorem is treated in all introductory texts on [[model theory]] or [[mathematical logic]].
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| === Historical publications ===
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| * {{Citation | last1=Löwenheim | first1=Leopold | author1-link=en:Leopold Löwenheim | title=Über Möglichkeiten im Relativkalkül | year=1915 | journal=Mathematische Annalen| issn=0025-5831 | volume=76 | pages=447–470 | doi=10.1007/BF01458217 | issue=4 | ref=harv }}
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| ** {{Citation|last1=Löwenheim|first1=Leopold|author1-link=Leopold Löwenheim|year=1977|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|chapter=On possibilities in the calculus of relatives|publisher=Harvard University Press|location=Cambridge, Massachusetts|edition=3rd|pages=228–251|isbn=0-674-32449-8|ref=harv}} ({{Google books|v4tBTBlU05sC|online copy|page=228}})
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| * {{Citation | last1=Maltsev | first1=Anatoly Ivanovich | author1-link=en:Anatoly Maltsev | title=Untersuchungen aus dem Gebiete der mathematischen Logik | year=1936 | journal=Matematicheskii Sbornik, n.s. | volume=1 | pages=323–336 | ref=harv }}
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| * {{Citation | last1=Skolem | first1=Thoralf | author1-link=Thoralf Skolem | title=Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen | year=1920 | journal=Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse | volume=6 | pages=1–36 | ref=harv }}
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| ** {{Citation|last1=Skolem|first1=Thoralf|author1-link=Thoralf Skolem|year=1977|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|chapter=Logico-combinatorical investigations in the satisfiability or provabilitiy of mathematical propositions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem|publisher=Harvard University Press|location=Cambridge, Massachusetts|edition=3rd|pages=252–263|isbn=0-674-32449-8|ref=harv}} ({{Google books|v4tBTBlU05sC|online copy|page=252}})
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| * {{Citation | author1-link=Thoralf Skolem | last1=Skolem | first1=Thoralf | year=1922 | title=Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre | journal=Mathematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse | pages=217–232 | ref=harv }}
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| ** {{Citation|last1=Skolem|first1=Thoralf|author1-link=Thoralf Skolem|year=1977|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|chapter=Some remarks on axiomatized set theory|publisher=Harvard University Press|location=Cambridge, Massachusetts|edition=3rd|pages=290–301|isbn=0-674-32449-8|ref=harv}} ({{Google books|v4tBTBlU05sC|online copy|page=290}})
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| * {{Citation | last1=Skolem | first1=Thoralf | author1-link=Thoralf Skolem | title=Über einige Grundlagenfragen der Mathematik | year=1929 | journal=Skrifter utgitt av det Norske Videnskaps-Akademi i Oslo, I. Matematisk-naturvidenskabelig Klasse | volume=7 | pages=1–49 | ref=harv }}
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| * {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | title=A System of Axioms for Geometry | doi=10.2307/1986462 | year=1904 | journal=Transactions of the American Mathematical Society| issn=0002-9947 | volume=5 | issue=3 | pages=343–384 | jstor=1986462 | ref=harv }}
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| | |
| === Secondary sources ===
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| * {{Citation | last1=Badesa | first1=Calixto | title=The Birth of Model Theory: Löwenheim's Theorem in the Frame of the Theory of Relatives | publisher=Princeton University Press| location=Princeton, NJ | isbn=978-0-691-05853-5 | year=2004 | ref=harv }}; A more concise account appears in chapter 9 of {{Citation|editor=Leila Haaparanta|title=The Development of Modern Logic|year=2009|publisher=Oxford University Press|isbn=978-0-19-513731-6 | ref=harv }}
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| * {{Citation|first1=Geraldine|last1=Brady|title=From Peirce to Skolem: A Neglected Chapter in the History of Logic|year=2000|publisher=Elsevier|isbn=978-0-444-50334-3 | ref=harv }}
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| * {{Citation | last1=Dawson | first1=John W., Jr. | title=The compactness of First-Order Logic: From Gödel to Lindström | year=1993 | journal=History and Philosophy of Logic | volume=14 | pages=15–37 | doi=10.1080/01445349308837208 | ref=harv }}
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| * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=Model theory | publisher=Cambridge Univ. Pr.| location=Cambridge | isbn=978-0-521-30442-9 | year=1993 | ref=harv }}
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| * {{Citation | last1=Poizat | first1=Bruno | title=A Course in Model Theory: An Introduction to Contemporary Mathematical Logic | publisher=Springer| location=Berlin, New York | isbn=978-0-387-98655-5 | year=2000 | ref=harv }}
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| == External links ==
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| * {{Mathworld|title=Löwenheim-Skolem Theorem|id=Loewenheim-SkolemTheorem|author=[[Alex Sakharov|Sakharov, Alex]] and [[Eric W. Weisstein|Weisstein, Eric W.]]}}
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| * Burris, Stanley N., [http://www.math.uwaterloo.ca/~snburris/htdocs/LOGIC/LOGICIANS/notes2.pdf Contributions of the Logicians, Part II, From Richard Dedekind to Gerhard Gentzen]
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| * Burris, Stanley N., [http://www.math.uwaterloo.ca/~snburris/htdocs/WWW/PDF/downward.pdf Downward Löwenheim–Skolem theorem]
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| * Simpson, Stephen G. (1998), [http://www.math.psu.edu/simpson/notes/master.pdf Model Theory]
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| {{logic}}
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| {{DEFAULTSORT:Lowenheim-Skolem Theorem}}
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| [[Category:Model theory]]
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| [[Category:Theorems in the foundations of mathematics]]
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| [[Category:Metatheorems]]
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to be able to eventually decay I think hundreds of years
Silent children, elders slowly opening day sentence: 'A treasure is to be attained become Dao, longevity Fam trivial matter experts to refine Tao, nor is it an easy thing Fortunately, your two pieces of magic. device,プラダ 財布 迷彩, the foundation has a very deep, but five prison Ong Teng function is too complex to excel Dao,プラダ 迷彩 財布, also must first gather eight million demons, then by hundreds of years of longevity Fam master fine blood exercise, to be able to eventually decay I think hundreds of years,プラダ スタッズ 財布, and so you may not be right. '
'He was right, five prison to become Tao Wang Ding,プラダ 財布 レディース, is not an easy thing.' Yan Fang cold in the body,プラダ最新財布, issued a voice.
'hundreds of years, really can not wait. Modao blood firmament that it?' Fang Han thought, is a fact.
'top quality Modao blood firmament is still five prison Wang Ding, only a single function to pure destruction known.'d better refining some, but I still want to spend a hundred years to
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seems to be well-intentioned
Green,prada 財布 通販, one of a witch word around his body suspended.
'big secret witch curse.'
side cold saw this old book that demon beneath books,prada 財布 スタッズ, old witch word emerged out of the book is a Mantra. Belong to something 巫道文 Ming.
'young, make haste to leave here not your place to belong elusive dòng days of restricted areas. after that there exists a wide variety of powerful, even feathering mén saints, but also not in depth which from the beginning to now, only 华天君 one day be able to smooth the elusive dòng being I was a kindness. '
that body withered,財布 プラダ, like the high priest of the old man saw the party generally cold, his eyes suddenly transform into kind, seems to be well-intentioned, good persuade parties to go cold.
'kindness,プラダ スタッズ 財布? give me out!'
side cold but does not see this old man, suddenly turned to look behind him.
Jie Jie,財布 プラダ レディース, Jie Jie
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その中に私と一緒に行く
市場のほとんどの縦断,prada 新作 財布。今、彼が言うすべてがそうしないと、理解の世界の外になり、私たちの小さな団体ではなく、定期的に礼拝に行く。 ' 「私は何かを購入したい！その中に私と一緒に行く,プラダ レディース 財布。 '
側風邪は僧侶が道の前で踊って、手を振った,prada ベルト。 目の
きらめきの間に、それは素晴らしい平野、新しい建物の要塞の裁判所の前の海の水平線の上に上陸した。この都市は、高尚なまでの紺のドレスを着て、生意気1僧侶立って、町の門で、山のように立っている。
「ストップ,プラダ最新財布！地平線シークラブの強豪を上陸させ、僧侶たちは歩くことになります。 '
低温側と光華、上空を飛ぶ重不滅フル10以上の長寿のマスターは、ドリンクを押し込んだ僧侶全体にこのクリーチャーを見ました。低温側と僧侶の道を生きて停止します,プラダ新作バッグ2014。 前任者 '
、怒らないで、古代の海の地平線地平線裁判所の命令は、統一された理解の世界を送信する
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オーダーの下
大規模な配列が競合の下に配置することができる,プラダ レディース 財布。 オーダーの下
Gutianquanメインドア、立ち上がる。
バン,prada スタッズ 財布！ ストリップ
、空隙に登場し、突然武器を飛び出す彼の手のフリックは、古代の名残マップ、大気上記道路、雲に凝縮される,プラダ 財布 スタッズ。大気の無数の先祖の神々は、上記の回転、息の「祖先」で、パッシングは低温側の心に達した,財布 プラダ。
牙冷たい驚き 'これは.......です'： '！。トップグレードセント' 製品中の
セント、あなたは精錬することができるように、大羅ジン罪の法律を必要とし、トップグレードのセントは、あなたが成功する精錬することできるようにするには、より強力な先祖のセントを必要としています,prada 財布 リボン。
側風邪は、トップグレードのセントを見たことがない。
さて、このトップグレードGutianquanメインドア雰囲気の法則Cnの「前駆」のうちセント祭り、祖先、天の空隙を引き裂くのは簡単、ライブ群衆の守護者。
「これは、
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ドアのために、光大のまぐさとして記述することができます
ドアのために、光大のまぐさとして記述することができます。
「あなたはいつも優しいをお楽しみいただけますことを保証するためにケースに私に従うならば低温側、私は弟子としてあなたを受け入れる,prada 新作 財布！私はケースを絶たない、美しい雲が、それらのすべては、比類のない美しさだ,財布 プラダ！ '終わりなき皇后はShouyi趙を入れて、9を見てYinjiuヤン2悪魔だけでなく、悪魔の神ではなく、条件のうち。 羽ヤシの扉を密閉し教えるhuanhangrnは今、ビッグFour'dは自分たちの生活マップを取得したい。それらの間に、明らかに仮止めではなく、1心の、自分の利益のために、それぞれ。
'ああ,prada ベルト？現在、私は自分たちの生活は比較的良い数字れる専用入れ、最終的に4があるのですか,財布 プラダ レディース？'牙がそのような状況を見て寒さ、助けることができなかったが、チャネル」ではなく4、互いにコンテストで、最も高強度をマナの方は私とどのように彼のドアすべてで避難上のいくつかのシニア姉妹見習い,プラダ人気財布？ ' 私たちは古いペーストいるとき
「ハッハッハハッハッハ！ 'ナインヤン悪魔突然の笑いは、低温側を指して、「この小さな男は、私たちはまだ本当に、あなたを実行する
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