Philipp Lücke

Definable pathological sets

Set-theoretic objects whose construction requires the Axiom of Choice are often referred to as pathological sets. For many types of pathological sets of real numbers, results from descriptive set theory can be used to show that these objects cannot be defined by simple formulas in second-order arithmetic. In this talk, I want to present results dealing with the set theoretic definability of pathological objects of higher cardinalities, focussing on long well-orderings and maximal almost disjoint families of subsets of uncountable cardinals. I will present results dealing with the following aspects of this topic: (i) the existence of such objects at ω1 in determinacy models, (ii) the Σ1-definability of these sets at ω1 in the presence of large cardinals, and (iii) the Σ1-definability of such objects above a measurable cardinal. This is joint work in progress with Sandra Müller (Vienna).