We examine a simple, accessible illustration of axiomatization in modern mathematics, starting in the historic basis of Euclidean geometry, and viewing its evolution through Hilbert’s formulation (following Mazur’s Axiomatic Reasoning). We then study set theory, laying first a foundation of logic, and concluding with perhaps the most prototypical and important example of axiomatization, the Zermelo-Fraenkel Choice Axioms.
The process of Axiomatization is minimalist. One starts with a collection of statements about the world, and to axiomatize them is to find a small number of basic assumptions (axioms) and logical inferences from which all these statements can be deduced. In this way, the potentially vast garden of statements can be known with just a handful of seeds.
This is a student-led colloquium led by Nathan Kruse under the supervision of T. Barrett.