 
Summary: The metamathematics of ergodic theory
Jeremy Avigad
Department of Philosophy and Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
Abstract
The metamathematical tradition, tracing back to Hilbert, employs syntactic mod
eling to study the methods of contemporary mathematics. A central goal has been,
in particular, to explore the extent to which infinitary methods can be understood
in computational or otherwise explicit terms. Ergodic theory provides rich oppor
tunities for such analysis. Although the field has its origins in seventeenth century
dynamics and nineteenth century statistical mechanics, it employs infinitary, non
constructive, and structural methods that are characteristically modern. At the
same time, computational concerns and recent applications to combinatorics and
number theory force us to reconsider the constructive character of the theory and its
methods. This paper surveys some recent contributions to the metamathematical
study of ergodic theory, focusing on the mean and pointwise ergodic theorems and
the Furstenberg structure theorem for measure preserving systems. In particular, I
characterize the extent to which these theorems are nonconstructive, and explain
how prooftheoretic methods can be used to locate their "constructive content."
