Cofinality spectrum theorems in model theory, set theory and general topology
Cofinality spectrum theorems in model theory, set theory and general topology
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模型论、集合论和一般拓扑中的共尾谱定理
DOI：
10.1090/jams830
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发表时间：
2012
影响因子：
3.9
通讯作者：
S. Shelah
中科院分区：
文献类型：
--
作者：
M. Malliaris;S. Shelah
我们联系并解决了两个在完全不同的领域中的长期悬而未决的问题：模型论问题，SOP2是否按Keisler阶极大，以及来自一般拓扑/集合论的问题，是否p=t，关于连续统的基数不变量的最古老的问题。我们通过展示这些问题可以转化为一个更基本的问题的实例来做到这一点，我们使用模型理论的方法来陈述和完全解决这些问题。关于余定谱问题S，我们实质上是指两个模型M1的数据，这些模型可能用扩展语言编写充分集合论的代码，以及定义M1中线性阶的一组著名的公式∆S。设T是S的“树顶”，是最小的正则基数λ，使得M1中的一组派生树有一个没有上界的严格递增λ-序列.设C(S，ts)是正则基数对(κ1，κ2)的集合，使得κ1≤κ2λ，“或等价，使得(ω，λ；)κ，κ/D不包含κ=cf(κ)≤λ，isλ+-Good)的(κ，κ)-割。我们得到了几个结果，特别是在非简单理论中存在一个最小Keisler类。
We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods. By a cofinality spectrum problem s we essentially mean the data of a pair of models M M1 which code sufficient set theory, possibly in an expanded language, along with a distinguished set of formulas ∆s which define linear orders in M1. Let ts, the “treetops” of s, be the smallest regular cardinal λ such that one of a set of derived trees in M1 has a strictly increasing λ-sequence with no upper bound. Let C(s, ts) be the set of pairs of regular cardinals (κ1, κ2) such that κ1 ≤ κ2 λ,” or what is equivalent, such that (ω,<)λ/D contains no (κ, κ)-cuts for κ = cf(κ) ≤ λ, is λ+-good. We obtain several consequences, notably existence of a minimum Keisler class among the non-simple theories.