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Preprint Number 492

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492. M. Malliaris and S. Shelah
Model-theoretic properties of ultrafilters built by independent families of functions

Submission date: 13 August 2012.


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 λ (i.e. a point after which all antichains of P(λ)/D have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, {thus} from saturating any non-low theory. The paper's three main constructions are as follows. First, we construct a regular filter D on λ so that any ultrafilter extending D fails to λ^+-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 on λ > κ which is λ-flexible but not κ^{++}-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that \lcf(\aleph_0) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory.

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Full text arXiv 1208.2579: pdf, ps.

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