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2021 | Book

Advances in Mathematical Logic

Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Kobe, Japan, September 2018, Selected, Revised Contributions

Editors: Prof. Toshiyasu Arai, Prof. Makoto Kikuchi, Satoru Kuroda, Prof. Mitsuhiro Okada, Dr. Teruyuki Yorioka

Publisher: Springer Nature Singapore

Book Series : Springer Proceedings in Mathematics & Statistics


About this book

​Gaisi Takeuti was one of the most brilliant, genius, and influential logicians of the 20th century. He was a long-time professor and professor emeritus of mathematics at the University of Illinois at Urbana-Champaign, USA, before he passed away on May 10, 2017, at the age of 91.

Takeuti was one of the founders of Proof Theory, a branch of mathematical logic that originated from Hilbert's program about the consistency of mathematics. Based on Gentzen's pioneering works of proof theory in the 1930s, he proposed a conjecture in 1953 concerning the essential nature of formal proofs of higher-order logic now known as Takeuti's fundamental conjecture and of which he gave a partial positive solution. His arguments on the conjecture and proof theory in general have had great influence on the later developments of mathematical logic, philosophy of mathematics, and applications of mathematical logic to theoretical computer science.

Takeuti's work ranged over the whole spectrum of mathematical logic, including set theory, computability theory, Boolean valued analysis, fuzzy logic, bounded arithmetic, and theoretical computer science. He wrote many monographs and textbooks both in English and in Japanese, and his monumental monograph Proof Theory, published in 1975, has long been a standard reference of proof theory. He had a wide range of interests covering virtually all areas of mathematics and extending to physics. His publications include many Japanese books for students and general readers about mathematical logic, mathematics in general, and connections between mathematics and physics, as well as many essays for Japanese science magazines.

This volume is a collection of papers based on the Symposium on Advances in Mathematical Logic 2018. The symposium was held September 18–20, 2018, at Kobe University, Japan, and was dedicated to the memory of Professor Gaisi Takeuti.

Table of Contents

Reflection Principles, Generic Large Cardinals, and the Continuum Problem
Strong reflection principles with the reflection cardinal \(\le \aleph _1\) or \({<}\,2^{\aleph _0}\) imply that the size of the continuum is either \(\aleph _1\) or \(\aleph _2\) or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
Sakaé Fuchino, André Ottenbreit Maschio Rodrigues
On Supercompactness of
This paper studies structural consequences of supercompactness of \(\omega _1\) under \(\mathsf {ZF}\). We show that the Axiom of Dependent Choice \((\mathsf {DC})\) follows from “\(\omega _1\) is supercompact”. “\(\omega _1\) is supercompact” also implies that \(\mathsf {AD}^+\), a strengthening of the Axiom of Determinacy \((\mathsf {AD})\), is equivalent to \(\mathsf {AD}_\mathbb {R}\). It is shown that “\(\omega _1\) is supercompact” does not imply \(\mathsf {AD}\). The most one can hope for is Suslin determinacy. We show that this follows from “\(\omega _1\) is supercompact” and Hod Pair Capturing \((\mathsf {HPC})\), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. “\(\omega _1\) is supercompact” on its own implies that every Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “\(\omega _1\) is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.
Daisuke Ikegami, Nam Trang
Interpolation Properties for Sacchetti’s Logics
In this paper we investigate interpolation properties for Sacchetti’s logics \({\mathbf {wGL}}_n\), which are fragments of Gödel-Löb logic \(\mathbf {GL}\). We prove the effective Lyndon interpolation property for Sacchetti’s logics \({\mathbf {wGL}}_n\) using Shamkanov’s circular proof argument.
Sohei Iwata
Rosser Provability and the Second Incompleteness Theorem
This paper is a continuation of Arai’s paper on derivability conditions for Rosser provability predicates. We investigate the limitations of the second incompleteness theorem by constructing three different Rosser provability predicates satisfying several derivability conditions.
Taishi Kurahashi
On Takeuti’s Early View of the Concept of Set
Gaisi Takeuti occasionally talked about the concept of set. Based on these remarks, we can easily see that Takeuti had an original view of the concept of set. However, Takeuti gave no systematic development of his own view of the concept of set in its entirety. In this paper, we try both to put together Takeuti’s remarks on the concept of set, most of which are currently available only in Japanese, and to reconstruct Takeuti’s view of the concept of set in a manner as systematic as possible. In particular, we will take a look at a relatively early period of his career, and we will focus first on Takeuti’s distinction between two concepts of set and second on how a Hilbert-Gentzen finitistic proof-theoretic analysis can clarify one of these concepts of set.
Hidenori Kurokawa
On Countable Stationary Towers
In this paper, we investigate properties of countable stationary towers. We derive the regularity properties of sets of reals in \(L(\mathbf{R})\) from some properties of countable stationary towers without explicit use of strong large cardinals such as Woodin cardinals. We also introduce the notion of semiprecipitousness and investigate its relation to precipitousness and presaturation of countable stationary towers. We show that precipitousness of countable stationary towers of weakly compact height implies the regularity properties of sets of reals in \(L(\mathbf{R})\).
Yo Matsubara, Toshimichi Usuba
Reforming Takeuti’s Quantum Set Theory to Satisfy de Morgan’s Laws
In 1981, Takeuti introduced set theory based on quantum logic by constructing a model analogous to Boolean-valued models for Boolean logic. He defined the quantum logical truth value for every sentence of set theory. He showed that equality axioms do not hold, while axioms of ZFC set theory hold if appropriately modified with the notion of commutators. Here, we consider the problem in Takeuti’s quantum set theory that De Morgan’s laws do not hold for bounded quantifiers. We construct a counter-example to De Morgan’s laws for bounded quantifiers in Takeuti’s quantum set theory. We redefine the truth value for the membership relation and bounded existential quantification to ensure that De Morgan’s laws hold. Then, we show that the truth value of every theorem of ZFC set theory is lower bounded by the commutator of constants therein as quantum transfer principle.
Masanao Ozawa
Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds
In this paper, without the axiom of choice, we show that if a certain downward Löwenheim–Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of \(\mathsf {ZFC}\).
Toshimichi Usuba
Irrational-Based Computability of Functions
We investigate a sort of a unifying theory of computability of real functions, continuous or discontinuous, called here “irrational-based” (IB-) computability. All the examples which are computable in our various theories presented previously are IB-computable. The basic requirements of computability, the sequential computability and the effective continuity, are defined relative to computable irrational real sequences. The family of IB-computable functions is closed under IB-effective convergence. In order to certify the fruitfulness of IB-computability, quite a number of examples are presented.
Mariko Yasugi, Yoshiki Tsujii, Takakazu Mori
“Gaisi Takeuti’s Finitist Standpoint” and Its Mathematical Embodiment
Gaisi Takeuti’s mathematical achievements in consistency proofs of subsystems of second order arithmetic were always lined by his thoughts on the finitist standpoint. Takeuti’s insistence on the finitist standpoint originates in the requirement from the consistency proof per se. In its essence, it is to show the termination of a decreasing sequence from an order structure used in a consistency proof as clearly as possible. Takeuti has left many writings on the finitist standpoint. Some of them will be introduced and closely studied. Then a way of giving a mathematical form to Takeuti’s finitist standpoint will be proposed. This is done by means of a semi-formal theory of generalized functionals. One can explicitly present a functional which, given any decreasing sequence from the order structure in concern, evaluates the terminating point of the sequence.
Mariko Yasugi
Properness Under Closed Forcing
For every uncountable regular \(\kappa \), we give two examples of proper posets which turn improper in some \(\kappa \)-closed forcing extension.
Yasuo Yoshinobu
Advances in Mathematical Logic
Prof. Toshiyasu Arai
Prof. Makoto Kikuchi
Satoru Kuroda
Prof. Mitsuhiro Okada
Dr. Teruyuki Yorioka
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Springer Nature Singapore
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