• Title, Summary, Keyword: incompleteness theorem

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The Philosophy of Limits: Between Mathematics and Philosophy (한계의 철학 : 수학과 철학 사이)

  • Park, Chang Kyun
    • Journal for History of Mathematics
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    • v.29 no.1
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    • pp.31-44
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    • 2016
  • This essay aims to suggest roughly the "philosophy of limits." The limits mainly refer to those of human experiences and rational thoughts. The philosophy of limits consist of three theses and two consequences(L, M). (1) The limits are necessarily supervenient in the course of searching knowledge. (2) The limits cannot be dissipated ultimately. (3) To recognize the limits is not only an intellectual recognition but also a beginning of whole personality's reaction. (L) It is a rational decision to accept the limits and leave the margins (yeoback/yeoheuck) rather than to try to remove them. (M) To leave the margins (yeoback/yeoheuck) is characteristic of being human, and enables one to harmoniously communicate with others. To justify the philosophy of limits, this essay examine the limits discussed in mathematics and philosophy: set theory, Godel's Incompleteness Theorem, Galois Theorem in mathematics; and Hume, Kant, Kierkegaard, and Wittgenstein in philosophy. I try to interpret consciousness of limits in various cultures. I claim that consciousness of the limits contribute to lucidity of human identity, communication between persons, stimulation of creative thinking.

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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G$\ddot{o}$del의 부완전성정리와 수학적 진리

  • 김용국;김빙남
    • Journal for History of Mathematics
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    • v.1 no.1
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    • pp.71-75
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    • 1984
  • Whether the complete Hilbert program could be carried out was rendered very doubtful by results due to Godel. These results may be roughly characterized as a demonstration that, in any system broad enough to contain all the formulas of a formalized elementary number theory, there exist formulas that neither can be proved nor disproved within the system. In this paper, Godel's incompleteness theorem is explained roughly moreover formul system and machines being refered, related to his theory.

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INCOMPLETENESS OF SPACE-TIME SUBMANIFOLD

  • Kim, Jong-Chul
    • Journal of the Korean Mathematical Society
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    • v.36 no.3
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    • pp.581-592
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    • 1999
  • Let M be a properly immersed timelike hypersurface of $\overline{M}$. If M is a diagonal type, M satisfies the generic condition under the certain conditions of the eigenvalues of the shape operator. Moreover, applying them to Raychaudhuri equation, we can show that M satisfies the generic condition. Thus, by these results, we establish the singularity theorem for M in $\overline{M}$.

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COSMOLOGY, EPISTEMOLOGY AND CHAOS

  • Unno, Wasaburo
    • Publications of The Korean Astronomical Society
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    • v.7 no.1
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    • pp.1-7
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    • 1992
  • We may consider the following three fundamental epistemological questions concerning cosmology. Can cosmology at last understand the origin of the universe? Can computers at last create? Can life be formed at last synthetically? These questions are in some sense related to the liar paradox containing the self-reference and, therefore. may not be answered by recursive processes in finite time. There are, however. various implications such that the chaos may break the trap of the self-reference paradox. In other words, Goedel's incompleteness theorem would not apply to chaos, even if the chaos can be generated by recursive processes. Internal relations among cosmology, epistemology and chaos must be investigated in greater detail.

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G$\ddot{o}$del's Critique of Turings Mechanism (튜링의 기계주의에 대한 괴델의 비평)

  • Hyun Woosik
    • Journal for History of Mathematics
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    • v.17 no.4
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    • pp.27-36
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    • 2004
  • This paper addresses G$\ddot{o}$del's critique of Turing's mechanism that a configuration of the Turing machine corresponds to each state of human mind. The first part gives a quick overview of Turing's analysis of cognition as computation and its variants. In the following part, we describe the concept of Turing machines, and the third part explains the computational limitations of Turing machines as a cognitive system. The fourth part demonstrates that Godel did not agree with Turing's argument, sometimes referred to as mechanism. Finally, we discuss an oracle Turing machine and its implications.

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