“Gaisi Takeuti’s Finitist Standpoint” and Its Mathematical Embodiment

Gaisi Takeuti’s mathematical achievements in consistency proofs of subsystems of second order arithmetic were always lined by his thoughts on the finitist standpoint. Takeuti’s insistence on the finitist standpoint originates in the requirement from the consistency proof per se. In its essence, it is to show the termination of a decreasing sequence from an order structure used in a consistency proof as clearly as possible. Takeuti has left many writings on the finitist standpoint. Some of them will be introduced and closely studied. Then a way of giving a mathematical form to Takeuti’s finitist standpoint will be proposed. This is done by means of a semi-formal theory of generalized functionals. One can explicitly present a functional which, given any decreasing sequence from the order structure in concern, evaluates the terminating point of the sequence.