For the first time in UMS history, the topic under discussion will be...mathematical logic. You've most probably heard the story before: just around the turn of the 20th century, Professor David Hilbert, logico-maniacal genius, foisted his famous program upon the world, bent on complete logical precision and axiomatic domination of all mathematics. If it weren't for the efforts of one Kurt Gödel, a quiet yet courageous Austrian, we would still be in the clutches of that formalist fascist. The formalist school took the standpoint that any mathematical theory can be turned into a symbolic language by equipping it with a set of axioms and a rule for deducing new theorems from these. One natural question to ask is, if a statement in a certain theory is designated as true, does there exist a proof for it? In this talk I will present a particular axiomatic system for elementary number theory called Z1. I will use the "decision problem for Z1" to investigate whether there exists an algorithm that determines the provability of theorems. I will then derive a version of Gödel's 1st incompleteness theorem for the system Z1, demonstrating that there exist propositions in Z1 which are neither provable nor unprovable within Z1.
This talk should be accessible to everyone.