We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.
"Approximate counting by hashing in bounded arithmetic." J. Symbolic Logic 74 (3) 829 - 860, September 2009. https://doi.org/10.2178/jsl/1245158087