Eccentric sets

This thesis is an anthology of papers that I have written during my time as a PhD student, split into three main chapters of mathematical content that each represent a facet of my research.
Chapter 3. From [KR24; Rya24c]. I begin with ZF set theory without the axiom of choice (whereas Chapters 4 and 5 use full ZFC). In this setting it is possible that there is a set X such that, for some ordinal α, there is a surjection X → α but no injection α → X. Such eccentric sets (for ‘tis their name) inform us about the structure of their universe of sets, and thus I present the Hartogs–Lindenbaum spectrum, a documentation of eccentricity within a universe. I show bounds for the Hartogs–Lindenbaum spectrum in models of SVC before constructing a model of ZF with maximal Hartogs–Lindenbaum spectrum (that is, the universe is as eccentric as possible).
Chapter 4. From [Rya24a]. Within this chapter I examine maximal θ-independent families (where θ is a cardinal): ‘large’ collections A ⊆ P(X) for some X such that, in some sense, the elements of A are ‘independent’ or ‘random’. While maximal ℵ0-independent families are guaranteed to exist by Zorn’s Lemma, this dramatically fails for θ > ℵ0, instead requiring the presence of large cardinals. I exhibit a method of constructing proper classes of maximal θ-independent families by forcing over models with large cardinals.
Chapter 5. From [Rya24b]. I end with inspiration from model theory. An important concept in model theory (and computer science, as it happens) is VC dimension, a measurement of ‘shattering’. VC dimension was originally defined in a finitary manner, as model theory is wont to do, so I extend the definition to allow nuanced infinite dimensions, presenting string dimension. This gives rise to ideals of low-dimensional sets, and I investigate the covering numbers of these ideals.