• Korean Journal of Logic
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• v.11 no.2
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• pp.1-56
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• 2008

This article intends to examine the widespread assumption, which has been uncritically accepted, that Zermelo simply adopted Hilbert's axiomatic method in his axiomatization of set theory. What is essential in that shared axiomatic method? And, exactly when was it established? By philosophical reflection on these questions, we are to uncover how Zermelo's thought and Hilbert's thought on the axiomatic method were developed interacting each other. As a consequence, we will note the possibility that Zermelo, in his early as well as late thought, had views about the axiomatic method entirely different from that of Hilbert. Such a result must have far-reaching implications to the history of set theory and the axiomatic method, thereby to the philosophy of mathematics in general.

• Korean Journal of Mathematics
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• v.14 no.1
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• pp.79-83
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• 2006

In this paper, we apply Zermelo's problem of navigation on Riemannian manifolds to Hermitian manifolds. Using a similar technique with which we define a Randers metric in a Finsler manifold by perturbing Riemannian metric with a vector field, we construct an $(a,b,f)$-metric in a Rizza manifold from a Hermitian metric and a given vector field.

• 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 Korea Institute of Information and Communication Engineering
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• v.17 no.1
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• pp.117-126
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• 2013

A ship is intended to reach a specified target point in the minimum-time when it travels with a constant speed through a region of strong currents and its heading angle is the control variable. This is called the Zermelo's navigation problem. Its approximate solution for the minimum-time control may be found using the calculus of variation. However, the accuracy of its approximate solution is not high since the solution is based on a table form of inverse relations for some complicated nonlinear equations. To enhance the accuracy, this paper employs the neural network to represent the inverse relation of the complicated nonlinear equations. The accurate minimum-time control is possible with the interpolation property of the neural network. Through the computer simulation study we have found that the proposed method is superior to the conventional ones.

• Journal for History of Mathematics
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• v.24 no.3
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• pp.59-76
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• 2011

Starting from a collection V as a model which satisfies the axioms of NBG, we call the elements of V as sets and the subcollections of V as classes. We reconstruct the G$\ddot{o}$del's proof of the consistency of GCH and AC with the axioms of Zermelo-Fraenkel set theory by using Mostowski-Shepherdson mapping theorem, reflection principles in Tarski-Vaught theorem and Montague-Levy theorem and the fact that NBG is a conservative extension of ZF.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.1
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• pp.9-11
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• 2023

Intuitionistic Zermelo-Fraenkel (IZF) set theory is a set theory without the axiom of choice and the law of excluded middle (LEM). The weak excluded middle law (WEM) states that ¬𝜑∨¬¬𝜑 for any formula 𝜑. In IZF we show that LEM is equivalent to WEM plus the condition that any set not equal to the empty set has an element.

• Korean Journal of Logic
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• v.19 no.1
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• pp.83-106
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• 2016

We compare the approaches to natural numbers and the induction principles in Frege's Grundgesetze and in systems thereafter. We start with an illustration of Frege's approach and then explain the use of induction principles in Zermelo-Fraenkel set theory and in modern type theories such as Calculus of Inductive Constructions. A comparison among the different approaches to induction principles is also given by analyzing them in respect of predicativity and impredicativity.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.831-850
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• 2023

Our long-standing Metatheorem in Ordered Fixed Point Theory is applied to some well-known order theoretic fixed point theorems. In the first half of this article, we introduce extended versions of the Zermelo fixed point theorem, Zorn's lemma, and the Caristi fixed point theorem based on the Brøndsted-Jachymski principle and our 2023 Metatheorem. We show some of their applications to other fixed point theorems or theorems on the existence of maximal elements in partially ordered sets. In the second half, we collect and improve order theoretic fixed point theorems in the collection of Howard-Rubin in 1991 and others. In fact, we improve or extend several ordering principles or fixed point theorems due to Brézis-Browder, Brøndsted, Knaster-Tarski, Tarski-Kantorovitch, Turinici, Granas-Horvath, Jachymski, and others.