Colleen Duffy : abstract

Mathematical Logic
Colleen Duffy
Department of Mathematics
University of St. Thomas
"The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it." Bertrand Russell

Logic is the science of reasoning. It is a wide and varied field of study that is intimately connected with a variety of areas in mathematics and philosophy including, but not limited to, set theory, algebra, epistemology, foundations of mathematics, and symbolic logic. Some of these connections will be discussed in the manner of the above quote. First it is worthwhile to start with the ontology of numbers, which leads to one possible definition of numbers. Transfinite and infinitesimal numbers will then be considered along with some questions that arise from their existence. This will mainly include the Axiom of Choice (AC) and the Continuum Hypothesis (CH) in Zermelo-Fraenkel (ZF) set theory, and will continue with further topics from there. For instance, when talking about the AC and CH, it is helpful to also introduce Lebesgue measure, Gödel's incompleteness theorem, Boolean algebra, and various paradoxes first. The purpose is to present some basic, important topics in mathematical logic from its beginnings until today.