Languages in which self reference is possible
Languages in which self reference is possible
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可以自引用的语言
DOI：
10.2307/2964058
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发表时间：
1957
影响因子：
0.6
通讯作者：
R. Smullyan
中科院分区：
文献类型：
--
作者：
R. Smullyan
本文探讨具有足够强度的语义系统S，使得对于在S中可定义的任何集合W（在一种将被精确界定的意义上），必定存在一个句子X，它在S中为真当且仅当它是W的一个元素。我们称这样的X为W的一个塔尔斯基句子。它是一个（在纯粹外延意义上）说自身在W中的句子。如果W是在某个句法系统C中不可证的所有表达式的集合，那么X就是哥德尔句子，它（在S中）为真当且仅当它（在C中）不可证。我们提供了一种构造这些句子的新方法，它产生结构特别简单的句子。该方法适用于多种系统，包括一种初等算术形式以及一些自应用的原句法系统。在应用于前者时，我们得到了一个定理的极其简单和直接的证明，这个定理本质上是塔尔斯基定理，即初等算术的真集合不是算术可定义的。我们方法的关键在于使用某个函数，即“范数”函数，它取代了对角函数的经典用法。为了给出范数函数的一个启发式概念，让我们将一个表达式E（非正式英语的）的范数定义为E后面跟着它自己的引号。
This paper treats of semantical systems S of sufficient strength so that for any set W definable in S (in a sense which will be made precise), there must exist a sentence X which is true in S if and only if it is an element of W. We call such an X a Tarski sentence for W. It is the sentence which (in a purely extensional sense) says of itself that it is in W. If W is the set of all expressions not provable in some syntactical system C, then X is the Gödel sentence which is true (in S) if and only if it is not provable (in C). We provide a novel method for the construction of these sentences, which yields sentences particularly simple in structure. The method is applicable to a variety of systems, including a form of elementary arithmetic, and some systems of protosyntax self applied. In application to the former, we obtain an extremely simple and direct proof of a theorem, which is essentially Tarski's theorem that the truth set of elementary arithmetic is not arithmetically definable. The crux of our method is in the use of a certain function, the ‘norm’ function, which replaces the classical use of the diagonal function. To give a heuristic idea of the norm function, let us define the norm of an expression E (of informal English) as E followed by its own quotation.