Undecidability in algebra - Ng, Kieran Elodie (Nanyang Technological University, 2024)
The thesis first discusses first-order model theory and its key concepts, such as compactness, categoricity, and quantifier elimination. These concepts provide a method to construct complete theories. Afterwards, we examine the notion of incompleteness and Gödel's Incompleteness Theorems. The techniques used by Gödel to produce his incompleteness results will be discussed, as well as the various consequences. In particular, we will look at two independence results in Peano Arithmetic, Goodstein’s Theorem, and the Paris-Harrington Principle. Finally, the thesis will also touch on the basics on computability theory. This includes the Church-Turing Thesis and the Halting Problem. An application of undecidability will also be examined, in the form of groups with unsolvable word problems.