Model-theoretic properties of ultrafilters built by independent families of functions

M. Malliaris, S. Shelah

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Abstract

Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on  l (i.e., a point after which all antichains of P(l)/D have cardinality less than l) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filterDon l so that any ultrafilter extendingDfails to l+-saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter onl > κ which is l-flexible but not κ++-good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that lcf(ℵ0) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.

Original languageEnglish
Pages (from-to)103-134
Number of pages32
JournalJournal of Symbolic Logic
Volume79
Issue number1
DOIs
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014, Association for Symbolic Logic

Keywords

  • Keisler’s order
  • Regular ultrafilters
  • Saturation of ultrapowers
  • Unstable model theory

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