In August I gave a talk during the III jornada conjuntística at the Universidad Nacional de Colombia on Finite combinatorics and logic. The talk was based on three papers. One of them, Goodstein’s function has been accepted for publication in the Revista Colombiana de Matemáticas. You can find it in my papers page, or see below.
A discussion of Goodstein’s function and issues related to independence in can be found in some of the posts in this blog marked with the category 117b: Computability theory (117b is the course on undecidability I taught at Caltech).