Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

Learning to teach database design by trial and error

Proceedings of: 4th International Conference on Enterprise Information Systems (ICEIS 2002), Ciudad Real, Spain, April 3-6, 2002The definition of effective pedagogical strategies for coaching and tutoring students according to their needs in each moment is a high handicap in ITS design. In this paper we propose the use of a Reinforcement Learning (RL) model, that allows the system to learn how to teach to each student individually, only based on the acquired experience with other learners with similar characteristics, like a human tutor does. This technique avoids to define the teaching strategies by learning action policies that define what, when and how to teach. The model is applied to a database design ITS system, used as an example to illustrate all the concepts managed in the model

A grammatical study of Ekpeye.

This thesis describes the grammatical structure of Ekpeye, an unwritten language of Eastern Nigeria. In the first chapter, there is an introduction to the Ekpeye language and people, and to the linguistic theory underlying this description, together with an outline of the analysis. Chapter 2 provides details of the transcription used in the thesis, and other phonological points. Chapters 3 to 13 contain the main body of the grammatical description, with units described in descending order of rank. The Sentence is outlined in chapter 3, and the Clause with its four classes and five types, in chapter 4, Chapters 5 to 7 deal with Phrase rank, a separate Phrase class being considered in each chapter. Chapter 5 contains the Nominal Phrase class with its five types, chapter 6 the Verbal Phrase class with its five types, and chapter 7 the Adverbial Phrase class with its single type. Chapters 8 to 11 describe Word rank in terms of four hyperclasses. Chapter 8 handles the Nominal hyperclass with its nine classes, and chapter 9 the Verbal hyperclass with its three classes and the three types found within one of the classes. Chapter 10 treats the Adverbial hyperclass with its three classes, and chapter 11 the Particle hyperclass with its eight classes. Stem rank is described in chapter 12, and Morpheme rank, with its two hyperclasses, in chapter 13. Chapter 14 contains an Ekpeye text fully analysed in accordance with the preceding description, and the thesis closes with a short bibliography

Semantic knowledge integration for learning from semantically imprecise data

Low availability of labeled training data often poses a fundamental limit to the accuracy of computer vision applications using machine learning methods. While these methods are improved continuously, e.g., through better neural network architectures, there cannot be a single methodical change that increases the accuracy on all possible tasks. This statement, known as the no free lunch theorem, suggests that we should consider aspects of machine learning other than learning algorithms for opportunities to escape the limits set by the available training data. In this thesis, we focus on two main aspects, namely the nature of the training data, where we introduce structure into the label set using concept hierarchies, and the learning paradigm, which we change in accordance with requirements of real-world applications as opposed to more academic setups.Concept hierarchies represent semantic relations, which are sets of statements such as "a bird is an animal." We propose a hierarchical classifier to integrate this domain knowledge in a pre-existing task, thereby increasing the information the classifier has access to. While the hierarchy's leaf nodes correspond to the original set of classes, the inner nodes are "new" concepts that do not exist in the original training data. However, we pose that such "imprecise" labels are valuable and should occur naturally, e.g., as an annotator's way of expressing their uncertainty. Furthermore, the increased number of concepts leads to more possible search terms when assembling a web-crawled dataset or using an image search. We propose CHILLAX, a method that learns from semantically imprecise training data, while still offering precise predictions to integrate seamlessly into a pre-existing application

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or Π11-CA—do not have least transitive models while weaker theories—from GBC to GBC + ETROrd —do have least transitive models

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of $T$-realizations of a fixed countable model of $\mathsf{ZFC}$, where $T$ is a reasonable second-order set theory such as $\mathsf{GBC}$ or $\mathsf{KM}$, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from $\mathsf{KM}$ to weaker theories. They showed that every model of $\mathsf{KM}$ plus the Class Collection schema "unrolls" to a model of $\mathsf{ZFC}^-$ with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as $\mathsf{GBC} + \mathsf{ETR}$. I also show that being $T$-realizable goes down to submodels for a broad selection of second-order set theories $T$. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from $\mathsf{GBC}$ to $\mathsf{KM}$. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories---such as $\mathsf{KM}$ or $\Pi^1_1\text{-}\mathsf{CA}$---do not have least transitive models while weaker theories---from $\mathsf{GBC}$ to $\mathsf{GBC} + \mathsf{ETR}_\mathrm{Ord}$---do have least transitive models.Comment: This is my PhD dissertatio