This is a continuation of the lectures getting the consistency of a weak threading ideal on \omega_2 from 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.

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.

For which pairs of linear orders $A$ and $B$ are the sums $A + B$ and $B + A$ isomorphic? In the 1930s, Tarski conjectured that $A + B = B + A$ if and only if

(i.) There is an order $C$ and natural numbers $n$ and $m$ such that $A = nC$ and $B = mC$, or

(ii.) There is an order $M$ such that $B = \omega A + M + \omega^* A$, or  

(iii.) There is an order $N$ such that $A = \omega B + N + \omega^* B$. 

Notably, these conditions on $A$ and $B$ are “arithmetic” in the sense that they are expressed in terms of finitary and $\omega$-ary sums of linear orders. 

Tarski proved his conjecture over the class of scattered linear orders, but Lindenbaum was able to produce a non-scattered counterexample. Building on Lindenbaum’s work, Aronszajn found a structural characterization of all additively commuting pairs of linear orders. 

Aronszajn’s characterization is somewhat complicated: in modern language it can be described in terms of orbit equivalence relations of groups of translations on $\mathbb{R}$. Tarski lamented that Aronszajn’s result could not be formulated arithmetically — that is, purely in terms of sums — and the line of work was abandoned. 

Building on our recent work on sums of linear orders, Eric Paul and I showed that there is an arithmetic condition equivalent to commutativity for linear orders. And in fact, the condition is a natural extension of the one appearing in Tarski’s original conjecture. In this talk, I will state our result, outline the proof, and discuss some related problems.