Computability and logic
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Some transfinite generalisations of gödel's incompleteness theorem
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
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Gödel's incompleteness theorem states that every finitely-presented, consistent, sound theory which is strong enough to include arithmetic is incomplete. In this paper we present elementary proofs for three axiomatic variants of Gödel's incompleteness theorem and we use them (a) to illustrate the idea that there is more than "complete vs. incomplete" there are degrees of incompleteness, and (b) to discuss the implications of incompleteness and computer-assisted proofs for Hilbert's Programme. We argue that the impossibility of carrying out Hilbert's Programme is a thesis and has a similar status to the Church-Turing thesis.