Counting as integration in feasible analysis

Abstract

Suppose that it is possible to integrate real functions over a weak base theory related to polynomial time computability. Does it follow that we can count? The answer seems to be: obviously yes! We try to convince the reader that the severe restrictions on induction in feasible theories preclude a straightforward answer. Nevertheless, a more sophisticated reflection does indeed show that the answer is affirmative.

The counting hierarchy in binary notation

“…The interest in systems connected with FCH instead of PTIME is justified by [6], where counting was proved to be a consequence of integration, even in weak systems like BTFA. Therefore, a theory like TCA 2 seems to be the appropriate setting to formalize Riemann Integration (see [7] and [8]).…”