We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf {IKP}$ and $\mathsf {CZF}$. Most previously studied large set axioms, notably, the constructive analogues of large cardinals below $0^\sharp $, have proof-theoretic strength weaker than full Second-Order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf {IKP}$ the basic properties of an elementary embedding $j\colon V\to M$ for $\Delta _0$-formulas, which we will denote by $\Delta _0\text {-}\mathsf {BTEE}_M$, we obtain the consistency of $\mathsf {ZFC}$ and more. We will also see that the consistency strength of a Reinhardt set exceeds that of $\mathsf {ZF+WA}$. Furthermore, we will define super Reinhardt sets and $\mathsf {TR}$, which is a constructive analogue of V being totally Reinhardt, and prove that their proof-theoretic strength exceeds that of $\mathsf {ZF}$ with choiceless large cardinals.

We give a short proof of a theorem of Beleznay asserting that the set $L2$ of reals coding linear orders of the form $I + I$ is complete analytic.

We give a simple proof that the first-order theory of well orders is axiomatized by transfinite induction, and that it is decidable.

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: [email protected]

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: [email protected].

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

The thesis uses various approaches to explore the algorithmic complexity of families of subsets of natural numbers. One of these approaches involves investigating upper semilattices of computable numberings of a given family and their complexity in different hierarchies. These semilattices, known as Rogers semilattices, can help distinguish different structural properties of families of partial computable functions and computably enumerable sets. As a result, by using Rogers semilattices of computable numberings, we can measure the algorithmic complexity of the corresponding family.

In the first part of thesis, we focus on limitwise monotonic numberings for families of limitwise monotonic sets and define their Rogers semilattices. The chapter investigates global invariants that show differences in the algebraic and elementary properties of the Rogers semilattices of families of sets from arithmetical hierarchy and Rogers semilattices of limitwise monotonic numberings. Such invariants include cardinality, laticeness, and types of isomorphism.

Within the second part of thesis, we explore the different forms of isomorphism exhibited by Rogers semilattices of families of sets in the analytical hierarchy. Additionally, we take into account various set-theoretic assumptions. Our research demonstrates that, when set-theoretic assumption known as Projective Determinacy is assumed, there exist an infinite number of non-isomorphic Rogers semilattices at each $\Sigma _n^1$-level of the analytical hierarchy.

I would also like to acknowledge the Grant of Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (AP19676989) and the Nazarbayev University Faculty Development Grant (N021220FD3851) for funding this research.

Abstract prepared by Zhansaya Tleuliyeva

[email protected].

URL: http://nur.nu.edu.kz/handle/123456789/7769.