Advances in Mathematical Logic: Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Kobe, Japan, September 2018, Selected, Revised

Preface
Organization
Contents
Reflection Principles, Generic Large Cardinals, and the Continuum Problem
	1 Gödel's Program and Large Cardinals
	2 Reflection Principles
	3 Game Reflection Principles and Generically Large Cardinals
	4 Simultaneous Reflection down to <20 and leq20
	5 Reflection Principles Under Large Continuum
	6 Laver-Generic Large Cardinals
	7 Some Open Problems and Takeuti's Account on Gödel's Contribution to the Continuum Problem
	References
On Supercompactness of ω1
	1 Introduction
	2 Definitions and Basic Concepts
	3 Choice Principles and Supercompactness of ω1
	4 Chang Model and Supercompactness of ω1
	5 Weak Homogeneity and Supercompactness of ω1
	6 AD+, ADmathbbR, and Supercompactness of ω1
	7 HPC and Supercompactness of ω1
	References
Interpolation Properties for Sacchetti's Logics
	1 Introduction
	2 Sacchetti's Logics wGLn
	3 Sequent Calculi for Sacchetti's Logics
		3.1 Kripke Semantics
		3.2 Proof Search Procedure
		3.3 Cut-Admissibility
	4 Craig Interpolation Property for wGLn
	5 Circular Proof System
	6 Lyndon Interpolation Property for wGLn
	7 Concluding Remarks
	References
Rosser Provability and the Second Incompleteness Theorem
	1 Introduction
	2 Provability Predicates and Derivability Conditions
	3 Rosser Provability Predicates
	4 Main Theorems
		4.1 Proof of Theorem 9
		4.2 Proof of Theorem 10
		4.3 Proof of Theorem 11
	References
On Takeuti's Early View of the Concept of Set
	1 Introduction
	2 An Overview of Takeuti's View of the Concept of Set
	3 Themes in Takeuti's View of the Concept of Set
		3.1 Critical Remarks on the Concept of Set
		3.2 The Concept of Set to Be Analyzed by Finitistic Methods
	4 Closing Remarks
	References
On Countable Stationary Towers
	1 Introduction
	2 Results
	References
Reforming Takeuti's Quantum Set Theory to Satisfy de Morgan's Laws
	1 Introduction
	2 Preliminaries
		2.1 Quantum Logic
		2.2 Commutators
		2.3 Conditionals
	3 Quantum Set Theory
	4 Transfer Principle
	References
Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds
	1 Introduction
	2 Preliminaries
	3 Löwenheim–Skolem Cardinals
	4 Uniform Definability of Grounds
	5 The Mantle and the Generic Mantle
	6 When AC is Forceable
	References
Irrational-Based Computability of Functions
	1 Introduction
	2 Preliminaries
	3 Irrational-Based Computable Function
	4 Examples
	5 Irrational-Based Effective Convergence of Function Sequence
	6 Final Note: Classical Account
	References
``Gaisi Takeuti's Finitist Standpoint'' and Its Mathematical Embodiment
	1 Introduction
	2 ``Takeuti's'' Hilbert's Program
	3 Takeuti's Finitist Standpoint
	4 My View
	5 Some Reflections on Takeuti's Thoughts
	6 Framework of Consistency Proofs
		6.1 Framework of Gentzen-Takeuti Consistency Proofs
		6.2 Proof-Theoretic Order Structure
	7 Accessibility Proofs
		7.1 Takeuti's Attempt: Eliminators
		7.2 Accessibility Proof of Ordinal Diagrams
		7.3 System ASOD
	8 Functional Interpretation
	9 Universe of Mechanisms
	10 Some Explanations
	11 In Retrospect
	References
Properness Under Closed Forcing
	1 Introduction
	2 The First Example
	3 The Second Example
	References