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Some Transfinite Generalisations of Gödel’s Incompleteness Theorem

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7160)

Abstract

Gödel’s incompleteness theorem can be seen as a limitation result of usual computing theory: it does not exist a (finite) software that takes as input a first order formula on the integers and decides (after a finite number of computations and always with a right answer) whether this formula is true or false. There are also many other limitations of usual computing theory that can be seen as generalisations of Gödel incompleteness theorem: for example the halting problem, Rice theorem, etc. In this paper, we will study what happens when we consider more powerful computing devices: these “transfinite devices” will be able to perform α classical computations and to use α bits of memory, where α is a fixed infinite cardinal. For example, \(\alpha = \aleph _0\,\) (the countable cardinal, i.e. the cardinal of ℕ), or \(\alpha =\mathfrak{C}\) (the cardinal of ℝ). We will see that for these “transfinite devices” almost all Gödel’s limitations results have relatively simple generalisations.

Keywords

  • Turing Machine
  • Limitation Result
  • Elementary Operation
  • Program Memory
  • Incompleteness Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Patarin, J. (2012). Some Transfinite Generalisations of Gödel’s Incompleteness Theorem. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_14

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27653-8

  • Online ISBN: 978-3-642-27654-5

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