Introduction to mathematical logic (3rd ed.)
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EUROCRYPT '02 Proceedings of the International Conference on the Theory and Applications of Cryptographic Techniques: Advances in Cryptology
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CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
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Mathematical Structures in Computer Science
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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.