Course overview -- Wintersemester 2022/2023 (Archive)

Positive logic is first-order logic where formulas are built without negation and using only existential quantifiers. By choosing the right languages to work with, this turns out to be a proper generalization of first-order logic. It is then natural to ask how much of usual model theory we can transfer to this setting; for example, one might ask about the dividing lines in classification theory, such as stability and simplicity: these notions, mostly introduced by Shelah, have been fruitfully used to classify theories with respect to various properties, for example the number of their models in a given cardinality or the existence of certain independence relations. In this talk I will briefly introduce positive model theory and some of the ideas about dividing lines, before discussing some work in progress (joint with Anna Dmitrieva and Mark Kamsma) about their interplay.

In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly applies to the theory of difference closed fields, which is simple and hence has possibly no saturated models in their uncountable cardinality.

Let GCH hold in $V$, and let $\kappa$ be a cardinal with a definable elementary embedding $j:V\rightarrow M$ such that ${\rm crit}(j)=\kappa$, ${}^{\kappa}M\subseteq M$ and $\kappa^{++}=(\kappa^{++})^{M}$ (in particular, $\kappa$ is measurable). H. Woodin proved that there is a cofinality preserving generic extension in which $\kappa$ stays measurable and GCH fails on it. This is achieved by using an Easton support iteration of Cohen forcings for having $2^{\alpha}=\alpha^{++}$ for every inaccessible $\alpha\leq\kappa$, and then adding an additional forcing to ensure the elementary embedding extends to the generic extension. Y. Ben Shalom proved in his thesis that this last forcing is unnecessary for the construction, and further extended the result to get $2^{\kappa}=\kappa^{+\gamma}$ assuming $\kappa^{+\gamma}=(\kappa^{+\gamma})^{M}$, for any successor ordinal $1<\gamma<\kappa$. We will present these results in some detail, and further extend the result of Ben Shalom for $\gamma=\kappa+1$ assuming $\kappa^{+\kappa+1}=(\kappa^{+\kappa+1})^{M}$.

An Ostaszewski club sequence is a weakening of Jensen's diamond. In contrast to the diamond, the club does not imply the continuum hypothesis. Numerous questions about the club stay open, and we know only few models in which there is just a club sequence but no diamond sequence. In recent joint work with Shelah we found that a winning strategy for the completeness player in a bounding game on a forcing order does not suffice to establish the club in the extension.

We use parametrized localized Ramsey spaces to define a new kind of forcing orders. There will be a generalized type of fusion sequence for showing that the forcings preserve $\aleph_1$.

The Halpern-Läuchli Theorem is a fundamental Ramsey type principle concerning partitions of finite products of trees. Historically the proof of the theorem was given using meta-mathematical reasoning. We will show a direct proof given by S.A Argyros, V. Felouzis and V. Kanellopoulos that uses anly standard mathematical arguments. The theorem talks about finite dimensional products of trees, but (time permitting) we will give a discussion of the infinite dimensional case.

Disjoint stationary sequences were introduced by Krueger to study forcings that add clubs through stationary sets. We answer a question of his by obtaining disjoint stationary sequences on successive cardinals. This talk will survey the area developed by Krueger and present the general idea of our new result.