It is widely acknowledged that Feferman maintained a sustained interest in predicativity throughout his academic career, while Takeuti diverged from this focus. Notably, Feferman delineated the limit of predicative reasoning during the 1960s. Additionally, he later proposed another approach related to predicativity called "unfolding."
On the other hand, Takeuti exhibited a strong interest in finitism, a divergence from Feferman’s intellectual path except for a single chapter in Proof Theory. Takeuti’s substantial contributions in developing a consistency program, rooted in his own finitist perspective, followed Gödel’s theorems.
This note is the first step in thorough investigation of the disparities between their respective philosophies concerning predicativity and finitism. An implication drawn from this inquiry is that these differences stem from a fundamental disparity in their foundational viewpoints in the context of the foundations mathematics. For Takeuti, the conception of finitism was still in an evolving state, necessitating further development and examination through his consistency program. In contrast, the conception of predicativism remained stable, with the understanding that the conception of predicatively definable sets was deemed as fixed.
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