On some early consistency proofs for fragments of arithmetic

I consider some early consistency proofs in the Hilbert School (by Ackermann, von Neumann and Herbrand) for fragments of arithmetic, before Goedel's results. The basic questions I shall try to answer are: What did they prove? Where (in what theory) can one prove what they proved? Which present-day formal theories can 'mimic' those contentual proofs, without violating Goedel's theorem on consistency proofs