### QUANTUM MECHANICS

, 2003

"... I consider in this book a formulation of Quantum Mechanics, which is often abbreviated as QM. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounte ..."

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I consider in this book a formulation of Quantum Mechanics, which is often abbreviated as QM. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio |x|/|v | between the space coordinate x and the velocity v, where |x|, etc. denotes the absolute value or length of a vector x. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system.

### Perspex Machine VII: The Universal Perspex Machine

"... with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the cont ..."

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with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of the paper are prohibited.

### KIMS-2003-07-07 Does Church-Kleene ordinal ωCK1 exist?

, 2003

"... Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ωCK1 really exists. We consider the systems S(α) defined in [2]. Let q̃(α) denote the Gödel number of Rosser formula or its negation A(α) ( = Aq(α)(q (α)) or ¬Aq(α)(q(α))), if the Rosser formula Aq(α)(q( ..."

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Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ωCK1 really exists. We consider the systems S(α) defined in [2]. Let q̃(α) denote the Gödel number of Rosser formula or its negation A(α) ( = Aq(α)(q (α)) or ¬Aq(α)(q(α))), if the Rosser formula Aq(α)(q(α)) is well-defined. By “recursive ordinals ” we mean those defined by Rogers [4]. Then that α is a recursive ordinal means that α < ωCK1, where ω CK 1 is the Church-Kleene ordinal. Lemma. The number q̃(α) is recursively defined for countable recursive ordinals α < ωCK1. Here ‘recursively defined ’ means that q̃(α) is defined inductively starting from 0.

### Does Church-Kleene ordinal ω CK 1 exist?

, 2003

"... Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ω CK 1 really exists. We consider the systems S (α) defined in [2]. Let ˜q(α) denote the Gödel number of Rosser formula or its negation A (α) ( = A q (α)(q (α) ) or ¬A q (α)(q (α))), if the Rosser formula ..."

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Abstract: A question is proposed if a nonrecursive ordinal, the so-called Church-Kleene ordinal ω CK 1 really exists. We consider the systems S (α) defined in [2]. Let ˜q(α) denote the Gödel number of Rosser formula or its negation A (α) ( = A q (α)(q (α) ) or ¬A q (α)(q (α))), if the Rosser formula A q (α)(q (α) ) is well-defined. By “recursive ordinals ” we mean those defined by Rogers [4]. Then that α is a recursive ordinal means that α < ω CK 1, where ω CK 1 is the Church-Kleene ordinal.