"Between Proof Theory and Model Theory"

ABSTRACT:

Proof theory, model theory, and algebraic logic are largely disjoint subjects, with independent goals and methods. In this talk, I will discuss some interesting points of overlap. First, I will discuss the general proof theoretic goal of proving *conservation theorems*, which provide a way of comparing axiomatic theories and measuring their strength. Then I will introduce a model-theoretic notion, that of an *Herbrand saturated model*, and show that this notion provides a smooth and uniform way of proving a number of important conservation results. In constrast to syntactic methods, the model-theoretic methods just mentioned are nonconstructive: they show that proofs in one theory can be translated to proofs in another, without providing an explicit translation. In the last part of this talk, I will show how algebraic forcing methods can be used to render the model-theoretic arguments constructive.
 

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