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Название: On Formally Undecidable Propositions of Principia Mathematica and Related Systems
Автор: Godel K.
It is very hard to find faults in what may be the most famous proof of the 20th century.For those not familiar with the Russell-Whitehead Principia Mathematica notationthis is a very hard book. I had the benefit of the Kac-Ulam explanation.I did find what might be problems with this proof.1) One is the reliance on number theory proofs about prime numbers that are assumed truein the G?delization of the primes coding of the mathematical axioms.2) The second is the assumption that the axioms statements represent the minimalrepresentation of such a system of axioms.Both are slim if none chances, but ones the G?del doesn't consider.Information theory was after this time where we discovered that a system of symbols can indeed at times be more efficiently coded.The best example of this seems to be Gray code compared to ordinary binary number code ( a number theory codelike G?del's prime code) where less turns out to be more in information terms.The theory of primes suffers from the new doctrine of strings that saysthat infinite scales don't exist in the "real" world: that a maximum and a minimumof measure are fixed parts of our reality. This kind of assumption can't be "proved"but is an axiom of a system of a mathematical sort and is counter to the Euclidean proof of an infinite number of primes.Primes already discovered by use of computers are much bigger than the numbers of ordinary physics, butwe are already reaching the Turing "stopping" problem in finding new ones.Some people equate in algorithmic information theory and number theorythe stopping problem with Infinity. That point of view of people like G. J. Chaitinis itself an unproved assumption. So the metamathematics used in the proof itself may be unprovable propositions.If so, then the proof based on such propositions can't itself be true.This argument in no way takes away from the greatness of G?del and his unique geniusas shown by this line of reasoning.