David Fernandéz-Duque
Université Paul Sabatier, Toulouse

Abstract: In this course we will give an introduction to ordinal notation systems based on collapsing cardinals in the style of Buchholz. Although familiarity with proof-theoretic ordinals is assumed, the course will work gradually, beginning with a quick review of predicative notation systems and introducing some intermediate systems along the way. The course will be divided in two sessions.

Session 1: November 2 (wednesday) 6pm - 7:30pm
We introduce a general notion of ordinal notation system and show how the usual presentations for $\varepsilon_0$ and $\Gamma_0$ fall into this framework. We discuss the behavior of an uncountable ordinal within a notation system and introduce collapsing functions, culminating in a notation system for the Bachmann-Howard ordinal $\psi(\varepsilon_{\Omega+1})$.

Session 2: November 3 (thursday) 3:30pm - 5pm
We establish some important properties of $\psi(\varepsilon_{\Omega+1})$, including its computability. We motivate the role of multiple cardinals by considering collapsing functions with two uncountable cardinals. We then show how these techniques are extended to collapsing infinitely many cardinals. Time permitting, we discuss how larger sets (e.g., an inaccessible) can be used to construct even larger proof-theoretic ordinals.

This seminar is supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013.