On Kreisel's explication of the concept of finitism
Finitistic methods appeared in the context of Hilbert's program in the foundations of mathematics. According to Hilbert, finitism contains those elementary methods of reasoning about finite objects (integers, words, ...) that are beyond any doubt and in some sense prior to any mathematical or axiomatic reasoning. Hilbert and Bernays gave examples of finitistically acceptable principles, but never clearly delineated the extent of these methods. Over the years, there were several proposals to make them mathematically precise, most notably by Kreisel and Tait.
This is a work in progress talk in which I will discuss Georg Kreisel's approach of characterizing finitism in terms of formal systems defined by iteration of certain processes of reflection. His answer is equivalent to the association of finitistic theorems with the (forall)(exists) theorems of Peano arithmetic. However, Kreisel's approach is technically involved and is itself in need of clarification. I will overview the ideas in Kreisel's paper and outline various associated problems.
Historically, the debate about the extent of finitism remained rather marginal; the issue can hardly ever be conclusively resolved. Nevertheless, the question is stubbornly alive, especially as a case study of the fundamental problem of how formal mathematical models of reasoning relate to contentual ones.
Lev D. Beklemishev graduated from the Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University in 1989. In 1989–1992 Ph.D. student; since 1992 researcher at V. A. Steklov Mathematical Institute of the Russian Academy of Sciences. Moscow Mathematical Society prize awarded in 1994 for the paper "On the classification of propositional provability logics". A. von Humboldt Fellowship (Muenster) 1998-99; Lise Meitner Fellowship (Vienna) 1999-2000. In 2000–2005 Universitair Docent at the Department of Philosophy of Utrecht University. In 2006 became a corresponding member, in 2019 academician of the Russian Academy of Sciences. Professor of the Faculty of Mathematics of the Higher School of Economics in Moscow. Chair of the Department of Mathematical Logic of Steklov Mathematical Institute.
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