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

Measures, slaloms, and forcing axioms

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