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Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic

Summary: Formalizing Forcing Arguments in Subsystems of
Second-Order Arithmetic
Jeremy Avigad
October 23, 1996
We show that certain model-theoretic forcing arguments involving sub-
systems of second-order arithmetic can be formalized in the base the-
ory, thereby converting them to effective proof-theoretic arguments. We
use this method to sharpen conservation theorems of Harrington and
Brown-Simpson, giving an effective proof that WKL+0 is conservative
over RCA0 with no significant increase in the lengths of proofs.
1 Introduction
Although forcing is usually considered to be a model-theoretic technique it has
a proof-theoretic side as well, in that forcing notions can usually be expressed
syntactically in the base theory. One can often use this fact to convert a model-
theoretic forcing argument to a proof-theoretic one, and in doing so obtain a
sharper and more effective version of the theorem being proven. We'll illustrate
this approach by formalizing two conservation results, one due to Harrington
and the other due to Brown and Simpson, involving subsystems of second-order
arithmetic. These results orginally appeared in the author's dissertation [1],


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


Collections: Multidisciplinary Databases and Resources; Mathematics