Based on the arithmetisation of language in Week 1 (which will be summarised) we give an overview of Gödel's incompleteness theorems, the Gödel-Rosser theorem, Tarski's Undefinability of truth, and the Arithmetised Completeness Theorem from the point of view of models of PA.
This is going to be a survey. For people familiar with these results from the proof theory literature, the model theoretic interpretations should be of interest. There will be plenty of "hints" why the results hold, but no detailed proofs. (Students following the seminary will be invited to attempt to write proofs of some these results, and if time permits perhaps present them later.)
Nonstandard finite groups are groups that occur as elements of a nonstandard model of PA. Viewed externally they are infinite, but the model of PA can prove most of the classical theory of finite groups. Sample results that were known before Reading's work include: a nonstandard finite torsion-free abelian group is divisible, and the alternating group A_n has no proper definable normal subgroups, but has infinitely many nondefinable ones, including a maximal normal subgroup.
Alan will present some of his own research into this topic.
The Midlands Logic Seminar was founded in 2011 and aims to cover all areas of mathematical logic, as well as related areas of theoretical computer science, and philosophy of mathematics. We typically have a study group session from 4:00-5:00 (term 1 topic: satisfcation classes) and a research talk from 5:00-6:00.
Dr Walter Dean
Department of Philosophy
University of Warwick