Set Theory and Algebra

1965 ◽  
pp. 1-52
Author(s):  
Edwin Hewitt ◽  
Karl R. Stromberg
Keyword(s):  
Set Theory ◽  
And Algebra

Set Theory and Algebra

1965 ◽  
pp. 1-52
Author(s):  
Edwin Hewitt ◽  
Karl Stromberg
Keyword(s):  
Set Theory ◽  
And Algebra

Set Theory and Algebra

1965 ◽  
pp. 1-52
Author(s):  
Edwin Hewitt ◽  
Karl Stromberg
Keyword(s):  
Set Theory ◽  
And Algebra

scholarly journals From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 397
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
World View ◽  
Open Problems ◽  
Boolean Valued Analysis ◽  
Mathematical World ◽  
Distributive Law ◽  
And Algebra ◽  
Adjoint Operators

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

scholarly journals Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras

1990 ◽  
Vol 117 ◽  
pp. 1-36 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Banach Space ◽  
Set Theory ◽  
Von Neumann Algebras ◽  
Space Theory ◽  
Von Neumann ◽  
Banach Modules ◽  
Space Objects ◽  
And Algebra ◽  
Module Structures ◽  
Transfer Principles

Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.

A Course on Set Theory

2009 ◽  
Author(s):  
Ernest Schimmerling
Keyword(s):  
Set Theory

Set Theory

2016 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Set Theory
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