  Tait’s technical analysis yields that the finitistic functions are exactly the primitive recursive ones, and the finitistic number-theoretic truths are exactly those provable in the theory of primitive recursive arithmetic \(\PRA\). It is important to stress that this analysis is not carried out from within the finitist viewpoint itself. Since the general notions of “function” and “proof” are not themselves finitary, the finitist is unable to make sense of Tait’s thesis that everything provable in \(\PRA\) is finitistically true. According to Tait, a proper analysis of finitistic provability must not assume that finitism itself has access to such non-finitistic notions. Kreisel’s approach and some criticisms of Tait that rely on reflection principles or \(\omega\)-rules run afoul of this requirement (see Tait 2002, 2005b). On the other hand, one could argue that \(\PRA\) is too strong a theory to count as a formalization of what is “presupposed by all non-trivial mathematical reasoning about numbers”: there are weaker but non-trivial theories which are related to smaller classes of functions than the primitive recursive ones, such as \(\PV\) and \(\EA\), related to the polynomial-time and Kalmar-elementary functions respectively (see Avigad 2003 on how much mathematics can be carried out in \(\EA)\). Using an analysis along the same lines as Tait’s,  Ganea (2010) has arrived at the  corresponding class of Kalmar-elementary functions as those that are finitistic.