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

by Malliaris and Shelah. [MiSh:997]
J Symbolic Logic, 2014
In this paper, the second of two, we continue our investigations of model-theoretic properties of ultrafilters: mu ({D}), the minimum size of a product of an unbounded sequence of natural numbers modulo {D} ; mathrm {lcf}(aleph_0, {D}) the lower cofinality (coinitiality) of aleph_0 modulo {D} ;flexibility, discussed extensively in the paper; realization of symmetric cuts; and goodness. We work in ZFC except where noted. Our main results are as follows. First, we prove that any ultrafilter {D} which is lambda-flexible (thus: lambda^+-o.k.) must have mu ({D}) = 2^lambda . Thus, a fortiori, {D} will saturate any stable theory. This is the strongest possible statement about the saturation power of flexibility alone, in light of our proof in the companion paper (I) that consistently, flexibility does not imply saturation of the random graph. In the remainder of the paper, we focus on the method of constructing ultrafilters via families of independent functions. Our second result is a constraint, that is, a tool for building ultrafilters which are not flexible. Specifically, we prove that if, at any point in a construction by independent functions the cardinality of the range of the remaining independent family is strictly smaller than the index set, then essentially no subsequent ultrafilter can be flexible. This is a useful point of leverage since any ultrafilter which is not flexible will fail to saturate any non-low theory. The third and fourth results are ultrafilter constructions. Third, assuming the existence of a measurable cardinal kappa (to obtain a kappa-complete ultrafilter), we prove that on any lambda >= kappa^+ there is a regular ultrafilter which is flexible but not good. This gives a second proof, of independent interest, of a question from Dow [1985], complementing the proof in the companion paper (I). Fourth, assuming the existence of a weakly compact cardinal kappa, we prove that for aleph_0 < theta = textrm {cf}(theta) < kappa <= lambda there is a regular ultrafilter {D} on lambda such that mathrm {lcf}(aleph_0, {D}) = theta but ({N}, <)^lambda / {D} has no (kappa, kappa)-cuts. This appears counter to model-theoretic intuition, since it shows some families of cuts in linear order can be realized without saturating the minimum unstable theory. We give several extensions of this last result, and show how to eliminate the large cardinal hypothesis in the case of asymmetric cuts.

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