Complete ineffability is a classical topic in Set Theory, much of it due to Baumgartner. A cardinal kappa being completely ineffable implies that kappa is inaccessible. Work of Eshkol, Foreman and Magidor has shown that ineffability is equivalent to the existence of certain threading ideals.
This talk describes a new kind of ideal: weakly threading ideals, and how to show that it is consistent that they exist on successor cardinals such as aleph_2 assuming the consistency of a measurable cardinal.
Complete ineffability is a classical topic in Set Theory, much of it due to Baumgartner. A cardinal kappa being completely ineffable implies that kappa is inaccessible. Work of Eshkol, Foreman and Magidor has shown that ineffability is equivalent to the existence of certain threading ideals.
This talk describes a new kind of ideal: weakly threading ideals, and how to show that it is consistent that they exist on successor cardinals such as aleph_2 assuming the consistency of a measurable cardinal.
Welch games are a genre of challenge-and-response games that can be used to stratify large cardinal strength between weak compactness and measurability [Foreman, Magidor, Zeman 2020]. The known variants of this game make sense only in the context of large cardinals. In this talk, we define and explore threading ideals, a combinatorial principle necessary for importing the Welch-game idea to successor cardinals.
Hardy fields are differential fields of (germs at infinity of) real-valued functions. Interest in them comes from several areas of mathematics, including asymptotic analysis, dynamical systems, and o-minimality. The first-order theory of existentially closed Hardy fields is completely axiomatizable and model complete in the language of ordered valued differential fields, as M. Aschenbrenner, L. van den Dries, and J. van der Hoeven have shown in a long and impressive series of works; in particular, all maximal Hardy fields are elementarily equivalent. Moreover, each maximal Hardy field can be equipped with an elementary differential subfield that is Dedekind complete in the maximal Hardy field. Along the lines of tame pairs of real closed fields (or tame pairs of o-minimal fields, more generally), the theory of such pairs is axiomatized by the notion of a transserial tame pair, the subject of this talk. After introducing these objects, I will summarize some of their properties. For example, they are model complete and topologically tame in the sense of being locally o-minimal and d-minimal, as well as satisfying a definable Baire Category Theorem.
Metric spaces and metric structures viewed from a model-theoretic perspective have attracted considerable attention in recent years. When the analogue of Scott's analysis is developed in the setting of continuous model theory, the rank of complete separable metric spaces (and structures) in continuous logic is always countable; this was done by Ben Yaacov, Doucha, Nies and Tsankov. An interesting problem arises if we equip a metric space with a natural, but classical, model-theoretic structure instead of a continuous logic structure. This situation was investigated by Fokina, Friedman, Koerwien and Nies (FFKN), and these authors asked if the Scott rank of complete, separable metric space in this way is always countable. In this talk I will give an example of a complete separable metric space which has Scott rank omega_1 when it is viewed as a classical model-theoretic structure as FFKN did. I will also say something about the proof, which is somewhat unusual because of it uses a fair amount of "serious" set theory.
