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Ryota Akiyoshi (Self and Infinity)

Takeuti on finite and infinite sets

by Ryota Akiyoshi (Keio Global Research Institute) and Andrew Arana (Université de Lorraine, Archives Poincaré)

Gaisi Takeuti is known best as a proof theorist, but he also wrote considerably on set theory. In his proof-theoretic writings on finitism, a novel view on the difference between finite and infinite sets emerges, in the context of qualifying epsilon_0 as a finitely accessible ordinal. In this talk I want to focus on this novel view, which I will show has a background in the Japanese philosophy of the Kyoto School of the early twentieth century, with which he was acquainted through various personal connections. Takeuti identifies a set with the kind of “mind” that generates it: finite minds, infinite minds, constructive minds, etc. A consequence of this view is that the boundary between the finite and infinite is more permeable than it is on the classical view coming from Cantor.