皆様，

東京工業大学の横山です． 下記の要領でセミナーを行います． どうぞご参加ください．

数理論理学セミナー (JSPS-FWO joint research seminar on mathematical logic)

日時：2月15日(金)14:30--18:00 (3講演) 場所：東京工業大学 大岡山西８号館W棟11階 W1101セミナー室

14:30--15:30 Sam Sanders (Ghent University, Munich Center for Mathematical Philosophy of LMU) Title: Reuniting the antipodes: bringing together nonstandard analysis and constructive analysis Abstract: Recently, Sanders introduced an interpretation of Errett Bishop’s Constructive Analysis (BISH) inside a particular system of classical Nonstan- dard Analysis called NSA. The role of ‘algorithm’ is played by the notion Ω-invariance; Intuitively, an object is Ω-invariant if it does not depend on the choice of infinitesimal used in its definition. The role of ‘proof’ is played by the Transfer Principle of Nonstandard Analysis as follows: In the same way as the Brouwer-Heyting-Kolmogorov-interpretation is limited to provable formulas, we only consider formulas A such that A ↔ ∗A in NSA, i.e. formulas which ‘satisfy Transfer’. As NSA does not include non-trivial Transfer Principles, only some formulas A satisfy A ↔ ∗A. This interpretation from BISH into Nonstandard Analysis can be called ‘natural’ and ‘faithful’ in the following threefold way: (i) Non-constructive principles (LPO, LLPO, MP, etc.) are interpreted as Transfer Principles which are not available in the system NSA. (ii) The interpretation preserves the equivalences of Constructive Reverse Mathematics. (iii) The interpretation preserves the property that the BISH-notion of algo- rithm is weaker than that of recursive function. We discuss the interpretation from BISH into NSA, and related topics.

15:45--16:45 Tin Lok Wong (Ghent University) Title: Countable numbers in a model of arithmetic: a survey Abstract: A *countable number* in a model M of Peano arithmetic is a number that has countably many predecessors in M. I will survey some results about these countable numbers as an initial segment of the model M. Most of this talk will be accessible to everyone who has some basic knowledge in cardinals and in the Completeness Theorem for first-order logic.

17:00--18:00 Florian Pelupessy (Ghent University) Title: Unprovable theorems. Abstract: We will provide a gentle introduction to some interesting natural theorems which are not provable in Peano Arithmetic. These theorems are natural in the sense that they are similar to ordinary theorems from the mathematics literature. We will discuss two important methods of showing unprovability for these theorems, one using results from proof theory and one from model theory.

------ 問い合わせ先：横山　啓太（東京工業大学） [email protected]

横山啓太