We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

Che 1-pu. Han'guk ŭi kaebyŏk -- 1. Kŭndae Han'guk konggongsŏng ŭi chŏn'gae wa t'aja waŭi yŏndae -- 2. Kŭndae Han'guk siminjŏk konggongsŏng ŭi sŏngnip -- 3. Taejonggyo pŏmt'unggusŭjuŭi wa pop'yŏnjuŭi -- Che 2-pu. Ilbon ŭi kaebyŏk -- 1. Kŭnse Ilbon sasang ŭi Sŏngin'gwan -- 2. Ilbon sinjonggyo ŭi kaebyŏk undong -- 3. Hyŏndae Ilbon ŭi saengmyŏng yŏngsŏng kwa ch'iyu yŏngsŏng -- Che 3-pu. Sirhak ŭi sigak -- 1. 19-segi sirhakcha ŭi Ilbon insik -- 2. Ch'oe Han-gi ŭi chonggyo hoet'ong sasang (...) -- 3. Han'guk, Ilbon, Chungguk e issŏsŏ 'Sinsirhangnon' pigyo -- Che 4-pu. Pigyo ŭi sigak -- 1. Ilbon esŏŭi T'oegye, Yulgok, Tasan yŏn'gu ŭi hŭrŭm -- 2. Ch'oe Han-gi wa Ilbon ŭi konggong sasangga pigyo yŏn'gu -- 3. Tongsŏyang konggongsŏng yŏn'gu wa Han'gukchŏk konggongsŏn. (shrink)

Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [7], Bull [1], [2], [3], Prawitz [6] etc. Here we treat modal logics based on the intuitionistic propositional logic, which call intuitionistic modal logics.

没後20年、いま必要な歴史に裏打ちされた予見力、現実政治に提言する率直さ、道義性と理想主義。.

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes $$\textrm{E}_k$$ and $$\textrm{U}_k$$ introduced in [1] are exactly the classes induced by $$\Sigma _k$$ and $$\Pi _k$$ respectively via the transformation procedure in (...) any first-order theory. (shrink)

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves (...) to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic. (shrink)

専門的な予備知識は仮定せずに完全性定理や計算可能性から論じ、第一および第二不完全性定理、Rosserの定理、Hilbertのプログラム、G ̈odelの加速定理、算術の超準モデル、Kolmogorov複雑性などを紹介して、不完全性定理の数学的意義と、その根源にある哲学的問題を説く。.

Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.