• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.25-29
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• 1995

The topos constructed in [6] is a set-like category that includes among its axioms an axiom of infinity and an axiom of choice. In its final form a topos is free from any such axioms. Set$\^$G/ is a topos whose object are G-set Ψ$\sub$s/:G${\times}$S\longrightarrowS and morphism f:S \longrightarrowT is an equivariants map. We already known that Set$\^$G/ satisfies the weak form of the axiom of choice but it does not satisfies the axiom of the choice.(omitted)

• The Pure and Applied Mathematics
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• v.10 no.3
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• pp.171-176
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• 2003

The purpose of this paper is to Show the relationships between three forms of the axiom of choice and well-ordering in an elementary topos.

• Journal for History of Mathematics
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• v.30 no.5
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• pp.261-287
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• 2017

De Zolt's axiom which is a precise formulation of Euclid's Common Notion 5, "the whole is greater than the part", for the notion of 'content' holds in any Hilbert plane. In this article, we study the history of de Zolt's axiom which has its origin in Euclid's Common Notions, and introduce an example of a plane geometry in which de Zolt's axiom does not hold. We show that there is no area function in this geometry and every square is equidecomposable with a square which is properly contained in the first one. From this we also show that there are two equidecomposable rectangles which have the same base and do not have the same altitude, and there is a rectangle which is equicomplementable with an emptyset.

• Journal of the Korean Society of Manufacturing Process Engineers
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• v.9 no.5
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• pp.7-13
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• 2010

Main starting valve is one of the main parts in the control system of diesel engines, purposed for starting main engines. It is composed of ball valve, check valve, actuator, etc. The design axioms provide a general framework for design methodology. Two axioms are independence axiom and information axiom. These axioms can be applied to all design process in a general way. The first axiom is introduced to analyze and evaluate the design of a main starting valve. The design parameters(DPS) are determined sequentially by considering the independence axiom. For the structural design of a main starting valve, the strength is calculated by using finite element method. In addition, the strength of its actuator piston is evaluated.

• Journal for History of Mathematics
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• v.32 no.5
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• pp.241-255
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• 2019

The proposition that the parallel axiom and the Pythagorean theorem are equivalent in the Hilbert geometry is true when the Archimedean axiom is assumed. In this article, we examine some specific plane geometries to see the existence of the non-archimidean Hilbert geometry in which the Pythagorean theorem holds but the parallel axiom does not. Furthermore we observe that the Pythagorean theorem is equivalent to the fact that the Hilbert geometry is actually a semi-Euclidean geometry.

• Korean Journal of Logic
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• v.20 no.1
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• pp.69-96
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• 2017

Wittgenstein criticizes explicitly Russell's theory of types and, in particular, his axiom of reducibility in the Tractatus Logico-Philosophicus. What, then, is the point of Wittgenstein's criticisms of Russell's theory of types? As a preliminary study to answer this question, I will examine how Wittgenstein criticized Russell's axiom of reducibility. Wittgenstein declares that Russell's axiom of reducibility is not a logical proposition, that if it is true it will be so mere by a happy chance and that "we can imagine a world in which the axiom of reducibility is not valid." What, then, is the ground for that? I will endeavor to show that by explicating the ideas of Wittgenstein's 1913 letter to Russell, those ideas decisively influenced on Ramsey's and Waismann's model which intended to show that the axiom of reducibility is not valid.

• Journal of the Korean Society of Manufacturing Technology Engineers
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• v.22 no.1
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• pp.100-106
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• 2013

This paper describes an integrated optimization design using multilevel decomposition technique on the base of the parametric distribution and independent axiom at the stages of lower level. Based on Pareto optimum solution, the detailed parameters at the lower level can be defined into the independent axiom. The suspension design is used as the simulation example.

• Journal for History of Mathematics
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• v.9 no.2
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• pp.1-9
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• 1996

This paper is a sequel to [26]. We investigate how the Axiom of Choice has been accepted after Zermelo introduced the Axiom in 1904. The response to the Axiom has divided into two groups of mathematicians, namely idealists and empiricists. We also investigate how the Zorn's lemma (1935) has been emerged. It was originally formulated by Hausdorff in 1909 and then by many other mathematicians independently.

• Journal of the Chungcheong Mathematical Society
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• v.16 no.2
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• pp.75-78
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• 2003

The Tychonoff product theorem is one of the most fundamental theorems in general topology. As is well-known, the proof of the Tychonoff product theorem relies on the axiom of choice. The converse was also conjectured by S. Kakutani and Kelley [1] then resolved this conjecture in his historical short note on 1950. However, the original proof due to Kelley has a flaw. According to this observation, we provide a correction of the proof in this paper.

• The Pure and Applied Mathematics
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• v.18 no.4
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• pp.353-359
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• 2011

In this paper, we characterize a semi-Riemannian manifolds satisfies the axiom of indefinite surfaces. We obtain the following result: If a semi-Riemannian manifold satisfies the axiom of indefinite surfaces, then it is a real space form.