Infinitary analogs of theorems from first order model theory
Infinitary analogs of theorems from first order model theory
复制标题
一阶模型理论定理的无限类比
DOI：
10.2307/2270256
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发表时间：
1971
影响因子：
0.6
通讯作者：
J. Malitz
中科院分区：
文献类型：
--
作者：
J. Malitz
这里介绍的材料属于Lκ，λ语言的模型理论。我们的结果要么是有限模型理论中重要定理的无限类比，要么表明这种类比是不存在的。例如，当i，和对i=1，2有相同的真Lω，ω句(即，初等等价)时，基数和1+2和+有相同的真Lω，ω句，直积1·2和·有相同的真Lω，ω句[3]。我们证明，当‘Lω，ω’被‘Lκ，λ’替换时，当且仅当κ强不可访问时，这是正确的。对于Lω1，ω解决了洛佩兹-埃斯科瓦尔提出的一个问题[7]。在§3中，我们完整地描述了Lκ，λ中的那些句子的表达能力，其中身份符号是唯一出现的关系符号。这推广了Hanf[4]的一个结果。
The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist. For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7]. In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].