Independence, order, and the interaction of ultrafilters and theories
The author considers the problem of realizing first-order types in regular ultrapowers. The author's previous work shows that it is enough to consider types determined by a single formula. In the article under review she investigates a class of formulas \(\varphi\) whose associated characteristic sequence of hypergraphs describes the realization of first-order and second-order types in ultrapowers, while at the same time pinpointing the properties of the corresponding ultrafilters.NEWLINENEWLINEThe author shows that each \(\varphi\) can be associated by means of its characteristic sequence to a ``second-order quantifier'' (in the sense of [\textit{S. Shelah}, Isr. J. Math. 15, 282--300 (1973; Zbl 0273.02009)]). The author goes on to show that many of Shelah's interpretability arguments go over to ultrapowers. As an application, she shows that any \(\varphi\) is dominated in Keisler's order by either the empty theory, the random graph, or by the minimal \(\mathrm{TP}_2\) theory. She also proves that the scope of the second-order quantifiers in Keisler's order does not go beyond \(\mathrm{TP}_2\). A final result indicates a possible gap in complexity between independence and strict order.
scientific article; zbMATH DE number 6086896
scientific article