Reading Kreisel: what we do (not) understand

This is a work in progress talk in which I will share my thoughts on Georg Kreisel's approach of characterizing finitism in terms of certain autonomous progressions of reflection principles. His well-known answer is the association of finitistic (in the sense of Hilbert) theorems with the Π2-theorems of Peano arithmetic. Rather than discussing/suggesting any alternative solutions to the problem of characterizing finitism (there are several other answers available), I will go over argumentation in Kreisel's paper [1] and outline what I do (not) understand there.