• Title, Summary, Keyword: $G{\ddot{o}}del$

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Gödel's Hermeneutics of the Relationship between Relativity Theory and Idealistic Philosophy (괴델이 해석하는 상대성이론과 관념론철학의 관계)

  • Hyun, Woosik
    • Journal for History of Mathematics
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    • v.27 no.1
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    • pp.59-66
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    • 2014
  • This interdisciplinary study explores G$\ddot{o}$del's hermeneutics of the relationship between relativity theory and idealistic philosophy in terms of time. For G$\ddot{o}$del, Einstein's contribution to the physical realization of idealistic philosophy would be remarkable. We start with a historical background around G$\ddot{o}$del's paper for Einstein(1949a). From the perspective of G$\ddot{o}$del's cosmology, the second part addresses the relative nature of time, and the next then investigates the rotating model of universes. G$\ddot{o}$del's own results show that the temporal conditions of relativity and idealistic philosophy are satisfiable in the mathematical model of rotating universes. Thus, it could be asserted to travel into any region of the past, present or future, and back again.

Mathematics as Syntax: Gödel's Critique and Carnap's Scientific Philosophy (구문론으로서의 수학: 괴델의 비판과 카르납의 과학적 철학)

  • Lee, Jeongmin
    • Korean Journal of Logic
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    • v.21 no.1
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    • pp.97-133
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    • 2018
  • In his unpublished article, "Is Mathematics Syntax of Language?," $G{\ddot{o}}del$ criticizes what he calls the 'syntactical interpretation' of mathematics by Carnap. Park, Chun, Awodey and Carus, Ricketts, and Tennant have all reconstructed $G{\ddot{o}}del^{\prime}s$ arguments in various ways and explored Carnap's possible responses. This paper first recreates $G{\ddot{o}}del$ and Carnap's debate about the nature of mathematics. After criticizing most existing reconstructions, I claim to make the following contributions. First, the 'language relativity' several scholars have attributed to Carnap is exaggerated. Rather, the essence of $G{\ddot{o}}del^{\prime}s$ critique is the applicability of mathematics and the argument based on 'expectability'. Thus, Carnap's response to $G{\ddot{o}}del$ must be found in how he saw the application of mathematics, especially its application to science. I argue that the 'correspondence principle' of Carnap, which has been overlooked in the existing discussions, plays a key role in the application of mathematics. Finally, the real implications of $G{\ddot{o}}del^{\prime}s$ incompleteness theorems - the inexhaustibility of mathematics - turn out to be what both $G{\ddot{o}}del$ and Carnap agree about.

Equivalence of Mind and Information Processing Formal System: $G{\ddot{o}}del's$ Disjunctive Conclusion and Incompleteness Theorems (마음과 정보처리형식체계의 논리적 동치성: 괴델의 선언결론과 불완전성 정리를 중심으로)

  • Hyun, Woo-Sik
    • Annual Conference on Human and Language Technology
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    • pp.258-263
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    • 1995
  • 마음과 기계의 관계에 대한 $G{\ddot{o}}del's$의 선언결론(disjunctive conclusion)은 마음과 정보처리형식체계의 논리적 동치성을 함의하고 있다. 그리고 $G{\ddot{o}}del's$의 불완전성 정리(Incompleteness Theorems)에 따르면 마음과 정보처리형식체계의 논리적 동치성은 무모순이며, 동치성 반증의 이론은 그 모델을 가질 수 없다.

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Gödel's Maximal Ontology (괴델의 극대 존재론)

  • Hyun, Woosik
    • Journal for History of Mathematics
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    • v.27 no.6
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    • pp.403-408
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    • 2014
  • The interdisciplinary study addresses the G$\ddot{o}$del's ontology from the perspective of the mathematical maximality. We first investigate G$\ddot{o}$del's God having all the positive properties as the intersection of ultrafilters in his own ontological proof(1970). Regarding the axiom of choice and his compactness theorem(1930), the next part discusses the ontological meaning of the maximal rather than the maximum in terms of an episteme space. The results show that G$\ddot{o}$del's ontological arguments imply all the existence of the maximal reality, and all the human's epistemological boundedness as well.

Frege and Gödel in Knowledge Change Model ('지식변화모델' 에서 프레게와 괴델)

  • Park, Chang Kyun
    • Journal for History of Mathematics
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    • v.27 no.1
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    • pp.47-57
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    • 2014
  • This paper aims to evaluate works of Frege and G$\ddot{o}$del, who play the trigger role in development of logic, by Knowledge Change Model. It identifies where their positions are in the model respectively. For this purpose I suggest types of knowledge change and their criteria for the evaluation. Knowledge change are classified into five types according to the degree of its change: improvement, weak glorious revolution, glorious revolution, strong glorious revolution, and total revolution. Criteria to evaluate the change are its contents, influence, pervasive effects, and so forth. The Knowledge Change Model consists of the types and the criteria. I argue that in the model Frege belongs to the total revolution and G$\ddot{o}$del to the weak glorious revolution. If we accept that the revolution in logic initiated by Frege was completed by G$\ddot{o}$del, it is a natural conclusion.

G$\ddot{o}$del's Mathematical Proof of the Existence of God (신의 존재에 대한 괴델의 수학적 증명)

  • Hyun, Woo-Sik
    • Journal for History of Mathematics
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    • v.23 no.1
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    • pp.79-88
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    • 2010
  • G$\ddot{o}$del's proof attempts to establish the existence of God by the definition that God is a being having all positive properties. The proof uses here second order modal logic system $S_5$ with the axiom ${\diamondsuit}{\Box}p{\rightarrow}{\Box}p$. We review the G$\ddot{o}$del's own version and prove his ontological theorems.

[ $G\ddot{o}del$ ] on the Foundations of Mathematics (괴델이 보는 수학의 토대)

  • Hyun, Woo-Sik
    • Journal for History of Mathematics
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    • v.20 no.3
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    • pp.17-26
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    • 2007
  • Following $G\ddot{o}del's$ own arguments, this paper explores his views on mathematics, its object, and mathematical intuition. The major claim is that we simply cannot classify the $G\ddot{o}del's$ view as robust Platonism or realism, since it is conceivable that both Platonistic ontology and intuitionistic epistemology occupy a central place in his philosophy and mathematics.

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On G$\ddot{o}$del′s Program from Incompleteness to Speed-up

  • 현우식
    • Journal for History of Mathematics
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    • v.15 no.3
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    • pp.75-82
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    • 2002
  • G$\ddot{o}$del's metamathematical program from Incompleteness to Speed-up theorems shows the necessity of ever higher systems beyond the fixed formal system and devises the relative consistency.

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Can Gödel's Incompleteness Theorem be a Ground for Dialetheism? (괴델의 불완전성 정리가 양진주의의 근거가 될 수 있는가?)

  • Choi, Seungrak
    • Korean Journal of Logic
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    • v.20 no.2
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    • pp.241-271
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    • 2017
  • Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest's argument for Dialetheism from $G{\ddot{o}}del^{\prime}s$ theorem is unconvincing as the lesson of $G{\ddot{o}}del^{\prime}s$ proof (or Rosser's proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest's inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying $G{\ddot{o}}del$ sentence to the inconsistent and complete theory of arithmetic. We argue, however, that the alternative argument raises a circularity problem. In sum, $G{\ddot{o}}del^{\prime}s$ and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of $G{\ddot{o}}del$ sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of $G{\ddot{o}}del$ sentence. Hence, $G{\ddot{o}}del^{\prime}s$ and its related theorem never can be a ground for Dialetheism.

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