A note on some early consistency proofs for fragments of arithmetic

I consider some early consistency proofs for fragments of arithmetic in the Hilbert School, before Goedel's results (Ackermann 1925, von Neumann 1927 and Herbrand 1931). My basic questions 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? I show that the case of Herbrand is especially interesting, since he explicitly tries to explain why Goedel's theorem on consistency proofs does not apply to his proof.