On languages with two variables
On languages with two variables
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DOI：
10.1002/malq.19750210118
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发表时间：
1975
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影响因子：
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通讯作者：
M. Mortimer
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文献类型：
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作者：
M. Mortimer
在逻辑事务的正常运行中，每种语言都包含无限多的变量。如果我们认为一种语言只有100多个变量，我们证明的能力和表达的能力就会受到严重的限制。HENKLX和MONK研究了前一类hnvc bwn的限制（见[2]和[3]）。我们在本文中关注的是后一种限制，特别是“无穷公理”的存在。只有无限模型的wntcnws。很容易看出，允许三个变量，这样的句子存在-例如，没有最后一个元素的严格线性排序的公理。iK siic* Jin n句子只使用一个变量，但带有函数符号。我们证明了这两个句子代表了可能的最佳结果，即不存在不带函数符号且使用只有无限模型的两个变量的一致sent-tw~。这回答了W.霍奇斯作为推论，我们得到了Scott的结果（~ cc（41）），即有两个变量而无函数符号的语言理论是可拓的。
In the normal run of logical affairs, every language contains infinitely many variables. SJmild we consider a language with only finitely many variables, our ability to prove, an (1 our ability to express become severely restricted. Restrictions of the former kind hnvc bwn studied by HENKLX and MONK (see [2] and [3]). Our concern in this paper i* with restrictions of the latter kind; specifically, the existence of “axioms of infinity” i. V. wntcnws with only infinite models. It is easily seen that, allowing tjhree variables,~ uch (t sentence exists-eg the axiom for a strict linear ordering without last element. iK siic* Ji n sentence using only one variable, but with function symbols. We prove that t hcw two sentences represent the best results possible-ie there is no consistent sent-tw~ without function symbols and using two variables which has only infinite models. This answers a question raised by W. HODGES. As a corollary we have SCOTT’S result (~ cc (41) that thc theory of a language wit, h two variables and no function symbols is dwidable.