When working in logic it is important to be clear about details. Importantly Franzén is pedantic in just the right measure. Franzén does a great job of trying to understand which parts of Chaitin's work really are significant. But a quick visit to his web site reveals that he has a slight problem with understanding the importance of his own work. Chaitin has done some great work, in particular he has a beautiful proof of Gödel's theorem using Kolmogorov complexity. He similarly dismisses some statements about the complexity of axiom systems made by Gregory Chaitin. So statements like "Godel's theorem demonstrates the existence of unprovable propositions" that abound in the literature are shown to be trivially false. For example if G is undecidable in formal system P then clearly it can be proved in P+G. Interestingly Dyson conceded this point when it was raised by Solomon Feferman in the New York Review of Books.įranzén makes some very simple statements that show the falsity of some of the claims made about the theorem. But the incompleteness might only be a feature of the arithmetic part of this system and have nothing to do with the physical part. Freeman Dyson has been using this to argue that physics is 'inexhaustible'. As a result the entire system will be incomplete. For example if we find a way to axiomatise physics it is likely we will be able to use some of those axioms for doing arithmetic. Additionally, Gödel's thoerem deals only with the arithmetic component of such systems. As Franzén points out, there are very complex systems that are incapable of being used for arithmetic and yet there are also some very simple systems that can be used for it. Gödel's theorem is explicitly about formal systems that are capable of formalizing 'part of arithmetic'. One of the ways in which the theorem is abused is the way it is applied to all kinds of formal and non-formal systems regardless of applicability. And I say 'abused' because all of the extra-mathematical accounts he quote are abuses. It looks closely at the various ways people have abused the theorem. And yet, it's hard to find work on Gödel's theorem that deals with these issues and yet is written by a logician with a good grasp of the subject. It has been invoked by theologists and literary critics.
Chaitin uses it as part of various discussions about irreducible complexity. Penrose uses it to argue for the specialness of human brains. Hofstadter makes it a central part of his discussion of AI in the classic Gödel, Escher, Bach.
Gödel's theorem has a habit of appearing in all sorts of literature. I just finished a first reading of Gödel's Theorem: An Incomplete Guide to its Use and Abuse by Torkel Franzén.
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