Abstract
let T be an ideal theory, by our lights ... Now T is consistent (by hypothesis) and has (only) infinite models. So by the completeness theorem ... T has a model of every infinite cardinality. Pick a model M of the same cardinality as THE WORLD. Map the individuals of M one-to-one into the pieces of THE WORLD, and use the mapping to define relations of M directly in THE WORLD. The result is a satisfaction relation SAT-a correspondence between the terms of L [that is, the first-order language in which T is expressed] and sets of pieces of THE WORLD-such that the theory T comes out tr-ue ... provided we just interpret true as TRUE(SAT). (Putnam 1977, pp. 125-6)
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