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.
A discussion of Goodstein’s function and issues related to independence in can be found in the page for my undecidability course in Caltech. I expect to keep updating that page from time to time.