a Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
b Computer Science Department, San Jose State University, One Washington Square, San Jose, CA 95192, USA
Abstract:
This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class PLS.