Some simple theories from a Boolean algebra point of view
ANNALS OF PURE AND APPLIED LOGIC(2024)
摘要
We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories Tm reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories Tn,k, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters "by hand" to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions. (c) 2023 Elsevier B.V. All rights reserved.
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关键词
Simple theories,Regular ultrafilters,Keisler's order,Saturation of ultrapowers
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