Matthias SCHIRN , Karl - Georg NIEBERGALL FINITISM = PRA ? ON A

tained that the chief difficulty for everyone who wishes to understand Hilbert’s conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning is essentially primitive recursive. In our paper, we attempt to show that his thesis “The finitist functions are precisely the primitive recursive functions” is disputable and that another, likewise defended by him, is untenable. The second thesis is that the finitist theorems are precisely those Π1-sentences that can be proved in (QF-IA). . 1 Tait’s interpretation of finitism In his influential essay ‘Finitism’ (1981), W.W. Tait set himself the task (a) of explicating the notion of finitism by explaining a sense in which one can prove general statements about the natural numbers without assuming