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, α = N 0 (the countable cardinal, i.e. the cardinal of ℕ), or α = C (the cardinal of ℝ). We will see that for these "transfinite devices" almost all Gödel's limitations results have relatively simple generalisations. © 2012 Springer-Verlag.
CITATION STYLE
Patarin, J. (2012). Some transfinite generalisations of Gödel’s incompleteness theorem. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7160 LNCS, pp. 183–193). https://doi.org/10.1007/978-3-642-27654-5_14
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