Note on a theorem of Putnam's

Michael Barr

In a 1981 book, H. Putnam claimed that in a pure relational language without equality, for any model of a relation that was neither empty nor full, there was another model that satisfies the same first order sentences. Ed Keenan observed that this was false for finite models since equality is a definable predicate in such cases. This note shows that Putnam's claim is true for infinite models, although it requires a more sophisticated proof than the one outlined by Putnam.