3.30 pm, Room 327, IMSc
Ordinal Analysis using Provability Logics
Alakh Dhruv Chopra
Chennai Mathematical Institute.
In recent years, there have been efforts to establish connections between proof-theoretic ordinal analysis, and provability logics and reflection principles. The main result in this area is the (first-order) analysis of Peano Arithmetic (PA) using a positive fragment of GLP (a polymodal extension of GL) and a careful stratification of PA in terms of consistency assertions. The central objects of concern are formulas expressing the aforementioned iterated consistency assertions -- referred to as worms -- and a defined well-ordering amongst them, which is used to extract the relevant ordinal notation system.
The talk will introduce the necessary logical framework: the positive fragment of GLP called Reflection Calculi (RC), worms, and their well-ordering; and use that to show a notation system for epsilon_0. A (very) brief sketch of the analysis of PA will also be given, in order to highlight a crucial step -- a so-called Reduction Property. Lastly, I will survey recent advances, in terms of both analyses and notation systems, and highlight open problems.
Only basic knowledge of mathematical logic, Peano Arithmetic, and ordinals would be assumed.
(This is Alakh's Masters' thesis presentation.)