Berkeley Cardinals and the search for V

This thesis is concerned with Berkeley Cardinals, very large cardinal axioms inconsistent with the Axiom of Choice. These notions have been recently introduced by J. Bagaria, P. Koellner and W. H. Woodin; our aim is to provide an introductory account of their features and of the motivations for investigating their consequences. As a noteworthy advance in the topic, we establish the independence from ZF of the cofinality of the least Berkeley cardinal, which is indeed the main point to focus on when dealing with the failure of Choice. We then explore the structural properties of the inner model L(V_\delta+1) under the assumption that delta is a singular limit of Berkeley cardinals each of which is a limit of extendible cardinals, lifting some of the theory of the axiom I_0 to the level of Berkeley cardinals. Finally, we discuss the role of Berkeley cardinals within the ultimate project of attaining a "definitive" description of the universe of set theory

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages

Defectiveness of formal concepts

It is often assumed that concepts from the formal sciences, such as mathematics and logic, have to be treated differently from concepts from non-formal sciences. This is especially relevant in cases of concept defectiveness, as in the empirical sciences defectiveness is an essential component of lager disruptive or transformative processes such as concept change or concept fragmentation. However, it is still unclear what role defectiveness plays for concepts in the formal sciences. On the one hand, a common view sees formal concepts to be protected against defects because of their precise and stable nature. On the other, studies going back as far as Lakatos (1963) showcase the changeability of such concepts. In this paper, I will investigate if and how defectiveness based on the occurrence of inconsistencies can appear with formal concepts. To make the case as strong as possible, I employ a strict notion of formal concept that assumes the concept to have a fixed and definite extension. I will show that there are indeed certain types of defectiveness that cannot occur with such concepts; but that there are other types of defectiveness that do occur. This means that while formal concepts have to be treated differently than non-formal concepts, questions about defectiveness---as raised in the conceptual engineering debate and philosophy of science---can still be applied to them. I will highlight this point by showing how formal sciences have special strategies available to them that allow them to resolve defectiveness of their concepts in a flexible and informative manner

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages

Large Cardinals in Weakened Axiomatic Theories

We study the notion of non-trivial elementary embeddings j from the set-theoretic universe, V, to itself under the assumption that V satisfies various classical and intuitionistic set theories. In particular, we investigate what consequences can be derived if V is only assumed to satisfy Kripke Platek set theory, set theory without Power Set or intuitionistic set theory. To do this, we construct the constructible universe in Intuitionistic Kripke Platek without Infinity and use this to find lower bounds for such embeddings. We then study the notion of definable embeddings before giving some initial bounds in terms of the standard large cardinal hierarchy. Finally, we give sufficient requirements for there to be no non-trivial elementary embedding of the universe in ZFC without Power Set. As a by-product of this analysis, we also study Collection Principles in ZFC without Power Set. This leads to models witnessing the failure of various Dependent Choice Principles and to the development of the theory of the Respected Model, a generalisation of symmetric submodels to the class forcing context