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Forcing in proof theory Jeremy Avigad

Summary: Forcing in proof theory
Jeremy Avigad
November 3, 2004
Paul Cohen's method of forcing, together with Saul Kripke's related
semantics for modal and intuitionistic logic, has had profound effects on
a number of branches of mathematical logic, from set theory and model
theory to constructive and categorical logic. Here, I argue that forcing
also has a place in traditional Hilbert-style proof theory, where the goal
is to formalize portions of ordinary mathematics in restricted axiomatic
theories, and study those theories in constructive or syntactic terms. I
will discuss the aspects of forcing that are useful in this respect, and some
sample applications. The latter include ways of obtaining conservation re-
sults for classical and intuitionistic theories, interpreting classical theories
in constructive ones, and constructivizing model-theoretic arguments.
1 Introduction
In 1963, Paul Cohen introduced the method of forcing to prove the indepen-
dence of both the axiom of choice and the continuum hypothesis from Zermelo-
Fraenkel set theory. It was not long before Saul Kripke noted a connection be-
tween forcing and his semantics for modal and intuitionistic logic, which had, in


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics