Some Transfinite Generalisations of Gödel’s Incompleteness Theorem

  • Jacques Patarin
Part of the Lecture Notes in Computer Science book series (LNCS, 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jacques Patarin
    • 1
  1. 1.University of VersaillesVersailles CedexFrance

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