Exact saturation in pseudo-elementary classes for simple and stable theories

被引:0
作者
Kaplan, Itay [1 ]
Ramsey, Nicholas [2 ]
Shelah, Saharon [1 ]
机构
[1] Hebrew Univ Jerusalem, Einstein Inst Math, IL-91904 Jerusalem, Israel
[2] Univ Notre Dame, Dept Math, South Bend, IN 46617 USA
基金
欧洲研究理事会; 以色列科学基金会;
关键词
Simple theories; stable theories; classification theory; exact saturation;
D O I
10.1142/S0219061322500209
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.
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页数:26
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