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This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.

The utilization of fuzzy mathematical modeling for quantification of the quality of training and educational delivery is an innovative application that can result in measurable and repeatable results. This research was designed to apply proven quantification techniques and Industrial Engineering methodologies to a nontraditional environment. The outcomes of this research provide the foundation, initial steps and preliminary validation for the development of a systematic fuzzy theoretical model to be applied for the quantification of various areas within training and education delivery. The test bed for this methodology is Orange County Public School system, the twelfth largest school district in the nation. The organizational and operational factors of a large school district are highly compatible with Systems Engineering concepts. The debate over education reform has drawn from referenced areas within the Industrial Engineering community including quality, continuous improvement, benchmarking and metrics development, data analysis, and scientific/systemic justification requirements. In spite of these applications, the literature does not reflect a consistent and broad application of these techniques in addressing the evaluation and quantification of educational delivery systems. This research draws on the previously listed areas within Industrial Engineering to apply these techniques to enhance the understanding and promote quantification of the multiple factors acting on the educational delivery system. The importance of addressing these issues is a national concern given the significant changes in the United States educational delivery system. For example, over the past 40 years there has been a more than three-hundred percent increase in per-pupil appropriations yet the academic performance gains have been limited and the quantification and measurement of those gains is even more limited. This body of work willidentify the systems, sub-systems, system factors, and factor degrees of existence necessary to quantify and measure these performance changes. Finally, the research will quantify the inputs and produce a model that provides a numeric value that represents the condition of the system and various subsystems of an educational system.

Yes / Process safety has drawn increasing attention in recent years and has been investigated from different perspectives, such as quantitative risk analysis, consequence modeling, and regulations. However, rare attempts have been made to focus on inherent safety design assessment, despite being the most cost-effective safety tactic and its vital role in sustainable development and safe operation of process infrastructure. Accordingly, the present research proposed a knowledge-driven model to assess inherent safety in process infrastructure under uncertainty. We first developed a holistic taxonomy of contributing factors into inherent safety design considering chemical, reaction, process, equipment, human factors, and organizational concerns associated with process plants. Then, we used subject matter experts, content validity ratio (CVR), and content validity index (CVI) to validate the taxonomy and data collection tools. We then employed a fuzzy inference system and the Extent Analysis (EA) method for knowledge acquisition under uncertainty. We tested the proposed model on a steam methane-reforming plant that produces hydrogen as renewable energy. The findings revealed the most contributing factors and indicators to improve the inherent safety design in the studied plant and effectively support the decision-making process to assign proper safety countermeasures.

This thesis explores Russell’s Paradox and the comparative analysis of Zermelo-Fraenkel set theory, von Neumann-Bernays-Gödel set theory, and Russell’s Type Theory from a mathematical Platonist perspective, focusing on the ontology of sets. Our conclusion posits that, although these theories have made significant attempts in addressing Russell’s paradox and other inconsistencies of naïve set theory, we currently lack a proper language for expressing set theory that fully captures the underlying Platonic world of sets. Consequently, it is impossible to definitively refute or accept any of the given theories as the ultimate solution the paradox.

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