 セミナー情報
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2014年 2月10日（月）～2月14日（金）
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| 2月14日(金) |
■ロジックセミナー 14:00-- 【会場：合同棟1201】 *開始時間が通常と異なります。
(1)14:00-15:00
未定
(2)15:10-16:10
講演者：Sam Sanders 氏（Ghent University）
題目： A new approach to Hilbert’s program via Nonstandard Analysis.
【概要】
David Hilbert’s program for the foundations of mathematics was an attempt at reducing infinitary mathematics to finitary mathematics,
as a way of establishing mathematics on firm and secure grounding, namely finitistic arithmetic. Thus, the original aim of proof theory was to
reduce all of mathematics to so-called finitistic arithmetic (later formalized in the system PRA), and to prove the consistency of finitistic mathematics
inside the latter. However, Kurt Goedel’s famous incompleteness theorems imply that such a consistency proof is impossible, i.e. that Hilbert program
cannot be accomplished in the aforementioned way. In this talk, we show how Nonstandard Analysis allows us to salvage Hilbert’s program; In particular,
we show how infinitary objects (from both classical and intuitionistic mathematics) can be reduced to nonstandard finitary objects in an elegant and
straightforward way. Ulrich Kohlenbach’s so-called higher-order Reverse Mathematics plays a central role.
(3)16:20-17:20
講演者：Paul Shafer 氏（Ghent University）
題目： Examples of problems that cannot be solved by randomness and examples of problems that can be solved by randomness
【概要】
The computational difficulty of a mathematical problem can be analyzed by considering the computable instances of the problem and determining how much extra information is needed to compute solutions to these instances. Often this can be accomplished by coding a well-known algorithmically unsolvable problem, such as the halting problem, into an instance of the problem being analyzed. However, many problems are weaker than the halting problem in the sense that such a coding is impossible.
In this talk, we consider several problems weaker than the halting problem and classify them according to whether or not access to randomness can help solve them.
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