Abstract. We discuss the notion of infinite exchangeability in the context of statistical mechanical systems. It turns out that many ferromagnetic systems (like Ising, Potts and Heisenberg models) can be extended to so-called infinite exchangeable processes. For the antiferromagnetic Ising model, we obtain sharp quantitative results on how much the system can be extended. One approach for doing this is via a solution of a new type of moment problem. This is joint work with Tom Liggett and Balint Toth.