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Is mathematics consistent?
, 2003
"... Abstract: A question is proposed whether or not set theory is consistent. We consider a formal set theory S, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of S, and will call it S (0). Definition 1. 1) ..."
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Abstract: A question is proposed whether or not set theory is consistent. We consider a formal set theory S, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of S, and will call it S (0). Definition 1. 1) We assume that a Gödel numbering of the system S (0) is given, and denote a formula with the Gödel number n by An. 2) A (0) (a, b) is a predicate meaning that “a is the Gödel number of a formula A with just one free variable (which we denote by A(a)), and b is the Gödel number of a proof of the formula A(a) in S (0), ” and B (0) (a, c) is a predicate meaning that “a is the Gödel number of a formula A(a), and c is the Gödel number of a proof of the formula ¬A(a) in S (0). ” Here a denotes the formal natural number corresponding to an intuitive natural number a of the meta level. Definition 2. Let P(x1, · · ·.xn) be an intuitivetheoretic predicate. We say that P(x1, · · ·,xn) is numeralwise expressible in the formal system S (0), if there is a formula P(x1, · · ·,xn) with no free variables other than the distinct variables x1, · · ·,xn such that, for each particular ntuple of natural numbers x1, · · ·,xn, the following holds: i) if P(x1, · · ·,xn) is true, then ⊢ P(x1, · · ·,xn). and ii) if P(x1, · · ·,xn) is false, then ⊢ ¬P(x1, · · ·,xn).
An implication of Gödel’s incompleteness theorem
, 2009
"... A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identif ..."
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A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication of these considerations is stated.
Does ChurchKleene ordinal ω CK 1 exist?
, 2003
"... Abstract: A question is proposed if a nonrecursive ordinal, the socalled ChurchKleene 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 socalled ChurchKleene 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 welldefined. 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 ChurchKleene ordinal.
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.
KIMS20030707 Does ChurchKleene ordinal ωCK1 exist?
, 2003
"... Abstract: A question is proposed if a nonrecursive ordinal, the socalled ChurchKleene 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 socalled ChurchKleene 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 welldefined. 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 ChurchKleene 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.