哥德尔、塔斯基、丘奇和骗子
Godel, Tarski, Church, and the Liar
摘要：
著名的哥德尔不完全性定理和所有逻辑悖论的原型--说谎者悖论--密切相关，这一事实当然不仅是众所周知的，而且是逻辑学家共同体的常识的一部分。事实上，几乎每一个或多或少的定理的正式处理（包括，就此而言，哥德尔的原始论文）都提到了这种联系。鉴于这种联系的存在是司空见惯的事实，更令人惊讶的是，人们对它的确切性质知之甚少。现在，从我们在这篇文章中所做的工作中可以看出，数学的三个中心极限定理，那些关于算术的不完备性和不可判定性以及其中真理的不可定义性的一般思想，可以被看作是解决说谎者悖论的不同方法。事实上，说谎者悖论的一个抽象的形式变体构成了一个一般的概念图式，揭示了它们共同的逻辑根源，将上面提到的定理联系起来，同时表明，在某种意义上，这些是唯一可能的相关限制定理，它们可以被认为是说谎者悖论的不同表现形式。另一方面，正如一个简单的例子所说明的那样，这个悖论的抽象版本开辟了一种可能性，即对真理和可证明性概念以外的概念形成相关的结果。
英文摘要：
The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community. Actually, almost every more or less formal treatment of the theorem (including, for that matter, Godel's original paper as well) makes a reference to this connection. In the light of the fact that the existence of this connection is a commonplace, all the more surprising that very little can be learnt about its exact nature. Now, it emerges from what we do in this paper that the general ideas underlying the three central limitation theorems of mathematics, those concerning the incompleteness and undecidability of arithmetic and the undefinability of truth within it can be taken as different ways to resolve the Liar paradox. In fact, an abstract formal variant of the Liar paradox constitutes a general conceptual schema that, revealing their common logical roots, connects the theorems referred to above and, at the same time, demonstrates that, in a sense, these are the only possible relevant limitation theorems formulated in terms of truth and provability alone that can be considered as different manifestations of the Liar paradox. On the other hand, as illustrated by a simple example, this abstract version of the paradox opens up the possibility to formulate related results concerning notions other than just those of the truth and provability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
