Summary: Saturated models of universal theories
A notion called Herbrand saturation is shown to provide the model-
theoretic analogue of a proof-theoretic method, Herbrand analysis, yield-
ing uniform model-theoretic proofs of a number of important conservation
theorems. A constructive, algebraic variation of the method is described,
providing yet a third approach, which is finitary but retains the semantic
flavor of the model-theoretic version.
Many important theorems in proof theory are conservation theorems, which is
to say, they have the following form: if a theory T1 proves a sentence of a
certain kind, then another theory T2 proves as well, or, perhaps, a specified
translation, . Typically the foundational interest in such a theorem lies in
the reduction of T1 to T2: though, on the surface, the principles embodied in
T1 may seem "stronger" or "more abstract" than those of T2, the conservation
theorem shows that there is at least a sense in which the stronger principles can
be eliminated in certain contexts.
Some examples of interesting conservation theorems are the following: