Tarski's undefinability theorem (original) (raw)
En logique mathématique, le théorème de Tarski, ou théorème de non définissabilité de Tarski, s'énonce informellement ainsi : On ne peut définir dans le langage de l'arithmétique la vérité des énoncés de ce langage.
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| dbo:abstract | Der Satz von Tarski über die Undefinierbarkeit der Wahrheit ist ein einschränkendes Ergebnis in der mathematischen Logik, das auf Alfred Tarski (1936) zurückgeht. Informell sagt der Satz, dass der Begriff der Wahrheit in einer Sprache nicht mit den Ausdrucksmitteln der Sprache selbst definiert werden kann. Die Beweisführung erfolgt über die sogenannten Tarski-Sätze, selbstreferenzielle Sätze der Form: ich bin ein Element von M für eine Menge M. Wählt man für M die Menge aller falschen Sätze eines Systems, führt die Konstruktion eines Tarski-Satzes zu einem Widerspruch: Ein wahrer Satz, der im System unbeweisbar ist. Daraus lässt sich folgern, dass die Menge aller wahren Sätze eines Systems nicht innerhalb dieses Systems definierbar ist. Dies ist kein Widerspruch zu den Beispielen formaler Systeme, die Tarski selbst angab, bei denen der Wahrheitsbegriff mit dem Beweisbarkeitsbegriff übereinstimmt. In diesen Fällen ist der Beweisbarkeitsbegriff nicht innerhalb des Systems definierbar. (de) En logique mathématique, le théorème de Tarski, ou théorème de non définissabilité de Tarski, s'énonce informellement ainsi : On ne peut définir dans le langage de l'arithmétique la vérité des énoncés de ce langage. (fr) Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. (en) Il teorema di indefinibilità di Tarski, enunciato e dimostrato da Alfred Tarski nel 1936, è un importante risultato limitativo della logica matematica, dei fondamenti della matematica e della semantica formale. L'enunciato si può esprimere, in termini non rigorosi, come La verità aritmetica non può essere definita all'interno dell'aritmetica. Il teorema si applica più generalmente ad ogni sistema formale sufficientemente potente, mostrando che la verità nel modello del sistema non può essere definita all'interno del sistema stesso. (it) 수리 논리학에서 타르스키의 정의 불가능성 정리(영어: Tarski's undefinability theorem)는 형식 의미론에 있어서 자기표현에 관한 중요한 제한을 가하는 정리이다. 이 정리를 비형식적으로 기술하면, "산술적 진리는 산술 내에서 정의될 수 없다"이다. 1936년 알프레트 타르스키가 기술하고 증명하였다. 이 정리는 충분히 강력한 모든 형식 체계(formal system)에 더욱 일반적으로 적용될 수 있는데, 이때는 "어떤 체계의 표준 모형 내에서의 진리는 그 체계 내에서는 정의될 수 없다"는 것을 보여준다. (ko) Teorema da indefinibilidade de Tarski, declarado e provado por Alfred Tarski em 1936, é um importante resultado limitativo em lógica matemática , os fundamentos da matemática, e em semântica formal. Informalmente, o teorema afirma que a verdade aritmética não pode ser definida em aritmética. O teorema se aplica de forma mais geral a qualquer sistema formal suficientemente forte, mostrando que a verdade no modelo padrão do sistema não pode ser definido dentro do sistema. (pt) Теорема Тарского о невыразимости арифметической истины — теорема, доказанная Альфредом Тарским в 1936 году, важный ограничивающий результат в математической логике, основаниях математики и формальной семантике. Теорема Тарского применима к любой достаточно сильной формальной системе. (ru) 塔斯基不可定義定理（英語：Tarski's undefinability theorem），是由阿爾弗雷德·塔斯基在1936年給出並證明，是在數理邏輯、數學基礎及形式化語義方面的一個重要的限制結果。簡單來說：我們無法在算術系統中定義何謂「算術的真理」。從而這個定理可被推廣成適用於任何足夠強的形式系統，以表明：我們無法在系統中定義何謂「系統標準模型的真理」。 (zh) |
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| rdfs:comment | En logique mathématique, le théorème de Tarski, ou théorème de non définissabilité de Tarski, s'énonce informellement ainsi : On ne peut définir dans le langage de l'arithmétique la vérité des énoncés de ce langage. (fr) Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. (en) Il teorema di indefinibilità di Tarski, enunciato e dimostrato da Alfred Tarski nel 1936, è un importante risultato limitativo della logica matematica, dei fondamenti della matematica e della semantica formale. L'enunciato si può esprimere, in termini non rigorosi, come La verità aritmetica non può essere definita all'interno dell'aritmetica. Il teorema si applica più generalmente ad ogni sistema formale sufficientemente potente, mostrando che la verità nel modello del sistema non può essere definita all'interno del sistema stesso. (it) 수리 논리학에서 타르스키의 정의 불가능성 정리(영어: Tarski's undefinability theorem)는 형식 의미론에 있어서 자기표현에 관한 중요한 제한을 가하는 정리이다. 이 정리를 비형식적으로 기술하면, "산술적 진리는 산술 내에서 정의될 수 없다"이다. 1936년 알프레트 타르스키가 기술하고 증명하였다. 이 정리는 충분히 강력한 모든 형식 체계(formal system)에 더욱 일반적으로 적용될 수 있는데, 이때는 "어떤 체계의 표준 모형 내에서의 진리는 그 체계 내에서는 정의될 수 없다"는 것을 보여준다. (ko) Teorema da indefinibilidade de Tarski, declarado e provado por Alfred Tarski em 1936, é um importante resultado limitativo em lógica matemática , os fundamentos da matemática, e em semântica formal. Informalmente, o teorema afirma que a verdade aritmética não pode ser definida em aritmética. O teorema se aplica de forma mais geral a qualquer sistema formal suficientemente forte, mostrando que a verdade no modelo padrão do sistema não pode ser definido dentro do sistema. (pt) Теорема Тарского о невыразимости арифметической истины — теорема, доказанная Альфредом Тарским в 1936 году, важный ограничивающий результат в математической логике, основаниях математики и формальной семантике. Теорема Тарского применима к любой достаточно сильной формальной системе. (ru) 塔斯基不可定義定理（英語：Tarski's undefinability theorem），是由阿爾弗雷德·塔斯基在1936年給出並證明，是在數理邏輯、數學基礎及形式化語義方面的一個重要的限制結果。簡單來說：我們無法在算術系統中定義何謂「算術的真理」。從而這個定理可被推廣成適用於任何足夠強的形式系統，以表明：我們無法在系統中定義何謂「系統標準模型的真理」。 (zh) Der Satz von Tarski über die Undefinierbarkeit der Wahrheit ist ein einschränkendes Ergebnis in der mathematischen Logik, das auf Alfred Tarski (1936) zurückgeht. Informell sagt der Satz, dass der Begriff der Wahrheit in einer Sprache nicht mit den Ausdrucksmitteln der Sprache selbst definiert werden kann. Die Beweisführung erfolgt über die sogenannten Tarski-Sätze, selbstreferenzielle Sätze der Form: ich bin ein Element von M für eine Menge M. Wählt man für M die Menge aller falschen Sätze eines Systems, führt die Konstruktion eines Tarski-Satzes zu einem Widerspruch: Ein wahrer Satz, der im System unbeweisbar ist. Daraus lässt sich folgern, dass die Menge aller wahren Sätze eines Systems nicht innerhalb dieses Systems definierbar ist. Dies ist kein Widerspruch zu den Beispielen formaler (de) |
| rdfs:label | Tarski's undefinability theorem (en) Tarskis Undefinierbarkeitssatz (de) Teorema di indefinibilità di Tarski (it) Théorème de Tarski (fr) タルスキの定義不可能性定理 (ja) 타르스키의 정의 불가능성 정리 (ko) Teorema da indefinibilidade de Tarski (pt) Теорема Тарского о невыразимости истины (ru) 塔斯基不可定義定理 (zh) |
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