We answer a question of Woodin [3] by showing that “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense” holds in a stationary set preserving extension of any universe with a cardinal $\kappa $ which is a limit of ${<}\kappa $-supercompact cardinals. We introduce a new forcing axiom $\mathrm {Q}$-Maximum, prove it consistent from a supercompact limit of supercompact cardinals, and show that it implies the version of Woodin’s $(*)$-axiom for $\mathbb Q_{\mathrm {max}}$. It follows that $\mathrm {Q}$-Maximum implies “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense.” Along the way we produce a number of other new instances of Asperó–Schindler’s $\mathrm {MM}^{++}\Rightarrow (*)$ (see [1]).

To force $\mathrm {Q}$-Maximum, we develop a method which allows for iterating $\omega _1$-preserving forcings which may destroy stationary sets, without collapsing $\omega _1$. We isolate a new regularity property for $\omega _1$-preserving forcings called respectfulness which lies at the heart of the resulting iteration theorem.

In the second part, we show that the $\kappa $-mantle, i.e., the intersection of all grounds which extend to V via forcing of size ${<}\kappa $, may fail to be a model of $\mathrm {AC}$ for various types of $\kappa $. Most importantly, it can be arranged that $\kappa $ is a Mahlo cardinal. This answers a question of Usuba [2].

Abstract prepared by Andreas Lietz

E-mail: andreas.lietz@tuwien.ac.at.

URL: https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf.

On May 23, 2024, the Fifth Section of the Court issued its judgment in Saakashvili v. Georgia concerning the immunity of former Georgian President Mikheil Saakashvili from prosecution for acts committed while in office. After giving up his Georgian citizenship and becoming a Ukrainian national, Saakashvili was convicted in absentia in two separate sets of criminal proceedings against him and was sentenced to a total of six years in prison. As part of a larger effort to make reparations for past wrongdoing, the Georgian Government received over 20,000 complaints from people claiming to be victims of serious human rights violations committed during the rule of Saakashvili's political party, the United National Movement, and under his presidency. The first case concerned a July 2005 attack on a member of parliament who was forced out of his car, beaten by several men, and was left permanently disfigured. The member of parliament alleged that the attack was retaliation for an interview he gave in which he spoke negatively about Saakashvili and his wife. The second case concerned Saakashvili's pardoning of four former high-ranking officials of the Ministry of the Interior who had been convicted of murder. That led to a separate investigation being opened in 2014 to explore charges of abuse of power.

This a reply to Emanuela Grama’s and Kevin Platt’s comments to my article “Emptiness Against Decolonization: Reflections from the Imperial Fault Line in eastern Latvia.”

In the Laws, Plato argues that legislation must not only compel, but also persuade. This is accomplished by prefacing laws with preludes. While this procedure is central to the legislative project of the dialogue, there is little interpretative agreement about the strategy of the preludes. This article defends an interpretation according to which the strategy is to engage with citizens in a way that anticipates their progress toward a more mature evaluative outlook, and helps them grow into it. The article shall refer to this strategy as proleptic engagement. While the virtuous ways of life required by law are intimately connected to happiness, the preludes do not persuade by spelling out this connection. Rather, they persuade by telling citizens what they need to hear so that they can come to appreciate this connection for themselves, in the context of their own lives. While the preludes are many and varied, this article argues that all preambular material can be understood as proleptic engagement.

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.

The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.

In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.

The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.

The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$“all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.

Abstract prepared by Clovis Hamel

E-mail: chamel@math.toronto.edu