Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Subtle and Ineffable Tree Properties

In the style of the tree property, we give combinatorial principles that capture the concepts of the so-called subtle and ineffable cardinals in such a way that they are also applicable to small cardinals. Building upon these principles we then develop a further one that even achieves this for supercompactness. We show the consistency of these principles starting from the corresponding large cardinals. Furthermore we show the equiconsistency for subtle and ineffable. For supercompactness, utilizing the failure of square we prove that the best currently known lower bounds for consistency strength in general can be applied. The main result of the thesis is the theorem that the Proper Forcing Axiom implies the principle corresponding to supercompactness.In Anlehnung an die Baumeigenschaft geben wir kombinatorische Prinzipien an, die die Konzepte der sogenannten subtle und ineffable Kardinalzahlen so einfangen, dass diese auch für kleine Kardinalzahlen anwendbar sind. Auf diesen Prinzipien aufbauend entwickeln wir dann ein weiteres, das dies sogar für superkompakte Kardinalzahlen leistet. Wir zeigen die Konsistenz dieser Prinzipien ausgehend von den jeweils entsprechenden großen Kardinalzahlen. Zudem zeigen wir die Äquikonsistenz für subtle und ineffable. Für Superkompaktheit beweisen wir durch das Fehlschlagen des Quadratprinzips, dass die besten derzeit bekannten unteren Schranken für Konsistenzstärke anwendbar sind. Das Hauptresultat der Arbeit ist das Ergebnis, dass das Proper Forcing Axiom das der Superkompaktheit entsprechende Prinzip impliziert

Combinatorics of countable ordinal topologies

We study combinatorial properties of ordinals under the order topology, focusing on the subspaces, partition properties and autohomeomorphism groups of countable ordinals. Our main results concern topological partition relations. Let n be a positive integer, let κ be a cardinal, and write [X] n for the set of subsets of X of size n. Given an ordinal β and ordinals αi for all i ∈ κ, write β →top (αi) n i∈κ to mean that for every function c : [β] n → κ (a colouring) there is some subspace X ⊆ β and some i ∈ κ such that X is homeomorphic to αi and [X] n ⊆ c −1 ({i}). We examine the cases n = 1 and n = 2, defining the topological pigeonhole number P top (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 1 i∈κ , and the topological Ramsey number Rtop (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 2 i∈κ . We resolve the case n = 1 by determining the topological pigeonhole number of an arbitrary sequence of ordinals, including an independence result for one class of cases. In the case n = 2, we prove a topological version of the Erd˝os–Milner theorem, namely that Rtop (α, k) is countable whenever α is countable and k is finite. More precisely, we prove that Rtop(ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and all positive integers k. We also provide more careful upper bounds for certain small ordinals, including Rtop(ω + 1, k + 1) = ω k + 1, Rtop(α, k) < ωω whenever α < ω2 , Rtop(ω 2 , k) ≤ ω ω and Rtop(ω 2 + 1, k + 2) ≤ ω ω·k + 1 for all positive integers k. Outside the partition calculus, we prove a topological analogue of Hausdorff’s theorem on scattered total orderings. This allows us to characterise countable subspaces of ordinals as the order topologies of countable scattered total orderings. As an application, we compute the number of subspaces of an ordinal up to homeomorphism. Finally, we study the group of autohomeomorphisms of ω n ·m+1 for finite n and m. We classify the normal subgroups contained in the pointwise stabiliser of the limit points. These subgroups fall naturally into D (n) disjoint sets, each either countable or of size 22 ℵ0 , where D (n) is the number of ⊆-antichains of P ({1, 2, . . . , n}). Our techniques span a variety of disciplines, including set theory, general topology and permutation group theor

The Universality Problem

The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof