Through a glass, plainly: is there a simple heuristic proof of Gödel's theorem? (original) (raw)

An Introduction to Gödel's Theorems

From the blurb: "In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book - extensively rewritten for its second edition - will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathem...

THOUGHT EXPERIMENT APPLIED TO GÖDEL'S INCOMPLETENESS THEOREM

This article offers a new reflection on the reasoning followed by Gödel in the proof of the incompleteness theorem(s). It takes the form of a short thought experiment using a variant of this theorem, obtained by cloning, and leading to an obviously unacceptable conclusion.

Experiments in Computational Metaphysics: Gödel's Proof of God's Existence

"Computer scientists prove the existence of God " - variants of this headline appeared in the international press in autumn 2013. Unfortunately, many media reports had only moderate success in communicating to the wider public what had actually been achieved and what not. This article outlines the main findings of the authors' joint work in computational metaphysics. More precisely, the article focuses on their computer-supported analysis of variants and recent emendations of Kurt Gödel's modern ontological argument for the existence of God. In the conducted experiments, automated theorem provers discovered some interesting and relevant facts.

How Gödel's Theorem Supports the Possibility of Machine Intelligence

Minds and Machines, 1998

Gödel’s Theorem is often used in arguments against machine intelligence, suggesting humans are not bound by the rules of any formal system. However, Gödelian arguments can be used to support AI, provided we extend our notion of computation to include devices incorporating random number generators. A com- plete description scheme can be given for integer functions, by which nonalgorithmic functions are shown to be partly random. Not being restricted to algorithms can be accounted for by the availability of an arbitrary random function. Humans, then, might not be rule-bound, but Gödelian arguments also suggest how the relevant sort of nonalgorithmicity may be trivially made available to machines.