Some Transfinite Generalisations of Gödel’s Incompleteness Theorem

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.