 
Summary: Forcing in proof theory
Jeremy Avigad
November 3, 2004
Abstract
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 Hilbertstyle 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 modeltheoretic 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
