Ordinal notations and well-orderings in bounded arithmetic

Ordinal notations and provability of well-foundedness have been a central tool in the study of the consistency strength and computational strength of formal theories of arithmetic. This development began with Gentzen’s consistency proof for Peano arithmetic based on the well-foundedness of ordinal notations up to ǫ0. Since the work of Gentzen, ordinal notations and provable wellfoundedness have been studied extensively for many other formal systems, some stronger and some weaker than Peano arithmetic. In the present paper, we investigate the provability and non-provability of well-foundedness of ordinal notations in very weak theories of bounded arithmetic, notably the theories S 2 and T i 2 with 1 ≤ i ≤ 2. We prove several results about the provability of well-foundedness for ordinal notations; our main results state that for the usual ordinal notations for ordinals below ǫ0 and Γ0, the theories T 1 2 and S 2 2 can prove the ordinal Σ1-minimization principle over a bounded domain. PLS is the class of functions computed by a polynomial local search to minimize a cost function. It is a corollary of our theorems that the cost function can be allowed to take on ordinal values below Γ0, without increasing the class PLS. The historical development of ordinal notations and formal theories of arithmetic is far too extensive for us to survey here. We shall include the basic definitions for ordinal notations of ordinals below ǫ0 and Γ0, and the reader can refer to Feferman [7, 8] or the textbooks of Schütte [14] or Pohlers [13] for more details. Theories of bounded arithmetic are fragments of Peano arithmetic which have induction strongly restricted, firstly to allow induction only on certain