Finite Axiomatizability using additional predicates
Finite Axiomatizability using additional predicates
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使用附加谓词的有限公理化
DOI：
10.2307/2964289
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发表时间：
1958
影响因子：
0.6
通讯作者：
R. Vaught
中科院分区：
文献类型：
--
作者：
W. Craig;R. Vaught
我们所说的理论总是指一阶理论，它有许多非逻辑常数。然后，对于有恒等式的理论（作为一个逻辑常数，在有恒等式的一阶逻辑中，理论在演绎下是封闭的），以及对于没有恒等式的理论，人们可以区分以下三个公理化的概念。第一，一个理论可以是递归可公理化的，或者，正如我们将说的，简单地说，是可公理化的。第二，一个理论可以通过使用额外的谓词而被公理化（f。a.+），由Kleene [9]引入的语法意义。最后，斜体字短语也可以从语义上解释。由此产生的概念将被称为s。F. a.+。它与Tarski [16]引入的模型论概念PC密切相关，或者更严格地说，与PC_AC_δ密切相关。对于有或没有恒等式的任意理论，很容易看出s。F. a.+包含f。a.+并且已知f. a.+意味着公理化。因此，很自然地要问，在什么条件下，匡威的含义成立，因为那时有关的概念是一致的，人们可以从一个概念转到另一个概念。Kleene [9]证明了：（1）对于任意无恒等式的理论，公理化蕴涵f。a.+。（2）对于只有无限模型的有恒等式的理论，公理化蕴涵f。a.+。
By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ. For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other. Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.
Lowenheim-Skolem-Tarski 的性质
DOI：
--
发表时间：
2022
期刊：
数理解析研究所考究録
影响因子：
--
作者：
Egashira Kento;Yata Kazuyoshi;Aoshima Makoto;Toshimichi Usuba;只木孝太郎;Kenta Ozeki;野崎寛;薄葉季路
通讯作者：
薄葉季路