Luke MacLean, Department of Pure Mathematics, University of Waterloo
"Kleene's $\mathcal{O}$"
A computable ordinal is one that corresponds to the order type of some computable linear well ordering. It can be shown that these are a subset of the countable ordinals, but there is still an issue on how to name these ordinals. We would like a system of notation that allows us to compare two arbitrary ordinals.
Enter Kleene's $\mathcal{O}$, a system of notation on ordinals and an ordering that, when restricted to the ordinals strictly below a fixed ordinal in the system, contains a unique notation for each such ordinal. We will prove some facts about this system and why it enables us to talk about computable ordinals. Furthermore we shall learn about the least non-computable ordinal $\omega_1^{CK}$ which is of interest because it is countable.
MC 5413