Godel Incompleteness Theorem

So what a skeptic may ask. Are the above examples manifestations of the computer s long-awaited, human-like intelligence No one familiar with Godel s theorem of incompleteness would ask such a question, (14,15) for this theorem states... 

In conclusion, we emphasise the following points (i) we have re-derived a previously obtained operator array formulation, which in its complex symmetric form permits a viable map of gravitational interactions within a combined quantum-classical structure (ii) the choice of representation allows the implementation of a global superposition principle valid both in the classical as well as the quantum domain (iii) the scope of the presentation has focused on obtaining well-known results of Einstein s theory of general relativity particularly in connection with the correct determination of the perihelion motion of the planet Mercury (iv) finally, we have obtained a surprising relation with Godel s celebrated incompleteness theorem. 

Suppose P to be false then P would have to be provable and hence true (to be consistent), a contradiction. Statement P must therefore be true but unprovable and this enabled Godel to prove his theorem. GOdel went on to show that any consistent and sufficiently rich mathematical system (i.e. containing at least arithmetic) will contain infinitely many statements which are true but not provable, and that one of them expresses the consistency of the system This is Godel s Second Incompleteness Theorem if a mathematical system is consistent then we cannot prove it to be so, by any proof which can be constructed within the system. 

Godel s incompleteness theorem (proved by Austrian-born American mathematician Kurt Godel) shows that it may not be possible to automatically prove an arbitrary theorem in systems as complex as the natural numbers. For simpler systems, such as group theory, automated theorem proving works if the user s computer can generate all reverse trees or a suitable subset of trees that can yield a... 

But some years ago, a mathematician named Godel devised a proof for a theorem that anything that is reasonably complex cannotenjoy this luxury (I believe he used the word interesting rather than reasonably complex). If your collection of information is factual, it cannot be entirely complete. And if it is complete, it cannot be entirely factual. In short, we will never know, we cannot ever know, every fact that constitutes an explanation of something. A complete book of knowledge must contain errors, and an error-free book of knowledge must be incomplete. 

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