Free groups and the axiom of choice

The Nielsen–Schreier theorem states that subgroups of free groups are free. As all of its proofs use the Axiom of Choice, it is natural to ask whether the theorem is equivalent to the Axiom of Choice. Other questions arise in this context, such as whether the same is true for free abelian groups, and whether free groups have a notion of dimension in the absence of Choice. In chapters 1 and 2 we define basic concepts and introduce Fraenkel–Mostowski models. In chapter 3 the notion of dimension in free groups is investigated. We prove, without using the full Axiom of Choice, that all bases of a free group have the same cardinality. In contrast, a closely related statement is shown to be equivalent to the Axiom of Choice. Schreier graphs are used to prove the Nielsen–Schreier theorem in chapter 4. For later reference, we also classify Schreier graphs of (normal) subgroups of free groups. Chapter 5 starts with an analysis of the use of the Axiom of Choice in the proof of the Nielsen–Schreier theorem. Then we introduce representative functions – a tool for constructing choice functions from bases. They are used to deduce the finite Axiom of Choice from Nielsen–Schreier, and to prove the equivalence of a strong form of Nielsen–Schreier and the Axiom of Choice. Using Fraenkel–Mostowski models, we show that Nielsen–Schreier cannot be deduced from the Boolean Prime Ideal Theorem. Chapter 6 explores properties of free abelian groups that are similar to those considered in chapter 5. However, the commutative setting requires new ideas and different proofs. Using representative functions, we deduce the Axiom of Choice for pairs from the abelian version of the Nielsen–Schreier theorem. This implication is shown to be strict by proving that it doesn’t follow from the Boolean Prime Ideal Theorem. We end with a section on potential applications to vector spaces

Current research on G\"odel's incompleteness theorems

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi

Set theory and the analyst

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure - category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know"

Second-order logic is logic

"Second-order logic" is the name given to a formal system. Some claim that the formal system is a logical system. Others claim that it is a mathematical system. In the thesis, I examine these claims in the light of some philosophical criteria which first motivated Frege in his logicist project. The criteria are that a logic should be universal, it should reflect our intuitive notion of logical validity, and it should be analytic. The analysis is interesting in two respects. One is conceptual: it gives us a purchase on where and how to draw a distinction between logic and other sciences. The other interest is historical: showing that second-order logic is a logical system according to the philosophical criteria mentioned above goes some way towards vindicating Frege's logicist project in a contemporary context

On the depth of G\"{o}del's incompleteness theorem

In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.Comment: 23 pages, revised version. arXiv admin note: text overlap with arXiv:2009.0488

Set Theory with Urelements

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails.Comment: arXiv admin note: text overlap with arXiv:2212.13627. Definition 15 in the previous versions is flawed, which is fixed in this versio

Large Cardinals

Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (V=L)