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Q: Mathematical Law ( Answered ,   1 Comment )
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 Subject: Mathematical Law Category: Science Asked by: tommykemper-ga List Price: \$5.00 Posted: 28 Mar 2003 12:33 PST Expires: 27 Apr 2003 13:33 PDT Question ID: 182464
 ```What is the mathematical theorem that states: "No theory can define a system more complex than itself?"```
 ```Hello Tommy, and thanks for a great question. I believe you are referring to the (nowadays) very famous logical proposition known as Godel's Incompletness Theorem (though I'm not sure that professional mathematicians would be completely comfortable with how you've paraphrased it in your question). Kurt Godel's Theorem, published in 1931, occupied a rather obscure position in the intellectual arena -- it was known to mathematicians, logicians and a few analytical philosphers, but not to too many others -- until the publication of Douglas Hofstadter's wonderful book, "Godel, Escher, Bach" in 1979. GEB, as it is affectionately known, is a very difficult read, but well worth it, if you're so inclined. It was described by one Hofstadter fan this way: "This book reads like an intellectual Grand Tour of hacker preoccupations. Music, mathematical logic, programming, speculations on the nature of intelligence, biology, and Zen are woven into a brilliant tapestry themed on the concept of encoded self-reference..." [ http://www.faqs.org/faqs/books/hofstadter-GEB-FAQ/ ] ------ That last little bit about "self reference" is the key, for Godel's Theorem says that every theory that refers to an element within a system will eventually create an element that *can't* be referred to from within the system. Huh? Perhaps this will help. It's an excerpt from a laymen's description of the Theorem, taken from Jones and Wilson's popular book, "An Incomplete Education": "Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules." The full description can be found at this site devoted to Godel's work: http://www.miskatonic.org/godel.html --------- In short, it's saying no matter how sophisticated your theory gets in describing a system, there's always a "bigger" system out there that just waiting to put your little theorem to shame....or something along those lines. ------- I love pontificating about things like this, so if you want any more information, explanation, B.S., whatever...just let me know through a Request for Clarification, and I'm happy to oblige. pafalafa-ga```
 ```The amazing thing about Goedel's Theorem from the standpoint of a mathematician is that he gives a purely constructive proof. That is, he actually writes down a real proposition in arithmetic that is formally undecidable, meaning that it cannot be shown to be true or false using arithmetic! And he shows that if you add that proposition to the axiom system of arithmetic, you can generate another such undecidable proposition. In fact, you can write down as many as you like. A book which takes you through the proof in easy stages is "Goedel's Proof" by Ernst Nagel and James R. Newman.```