  In this paper we present a proof of Goodman's Theorem, a classical result in
the metamathematics of constructivism, which states that the addition of the
axiom of choice to Heyting arithmetic in finite types does not increase the
collection of provable arithmetical sentences. Our proof relies on several
ideas from earlier proofs by other authors, but adds some new ones as well. In
particular, we show how a recent paper by Jaap van Oosten can be used to
simplify a key step in the proof. We have also included an interesting
corollary for classical systems pointed out to us by Ulrich Kohlenbach.
