Constructing κ‐like Models of Arithmetic
Constructing κ‐like Models of Arithmetic
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构建 κ 类算术模型
DOI：
10.1112/s002461079600470x
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
R. Kaye
中科院分区：
文献类型：
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作者：
R. Kaye
一个模型（M，<，.）是类κ的，如果M有基数κ，但对所有α ∈ M，{x ∈ M：x < a}的基数严格小于κ。在本文中，我们将给出对各种奇异基数κ，满足PA的任意大的有限部分但不满足PA本身的算术的κ-like模型的构造。主要结果如下：（1）对于每个可数的非标准M <$<$2 −Th（PA），其任意大的初始段满足PA，且每个不可数的共尾性为ω的κ，存在M的共尾扩张K，它是κ-like的;对于<$n−Th（PA），也是这个结果的分层变体;（2）对于每个n <$1，每个奇异κ和每个M <$B∑n+exp+<$I∑n，存在一个与M初等等价的κ-like模型K。
A model (M, <, …) is κ‐like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M : x < a} is strictly less than κ. In this paper we shall give constructions of κ‐like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M ⊧ Π2−Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ‐like; also hierarchical variants of this result for Πn−Th(PA); and (2) for every n ⩾ 1, every singular κ and every M ⊧ B∑n+exp+¬ I∑n there is a κ‐like model K elementarily equivalent to M.