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 ε_0 . Since the work of Gentzen, ordinal notations and provable well-foundedness 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^i_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 ε_0 and Γ_0, the theories T^i2 and S^2_2 can prove the ordinal Σ^b_1 – 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 Γ_0 , without increasing the class PLS .