Godel's Last Theorem

Discussion in 'Politics' started by BenTheMan, Mar 27, 2007.

  1. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    My office mate and I were talking about Goedel, and he told me this story that I had not known.

    The long and short of it is that Goedel found a technical flaw in the US Constitution that allowed for the creation of a dictatorship. Goedel never told anyone about this (at the insistence of Einstein, who told him that bringing this to the attention may jeoporadize his chances at citizenship).

    Has anyone else heard of this story?

    http://fourteenthcircuit.blogspot.com/fourteenthcircuit_archive.html

    I haven't any indications online that anyone ever found out what Goedel was thinking, but the tenth ammendment does say this:

    To the mods: Please move this if you think it needs to be moved.
     
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  3. Absane Rocket Surgeon Valued Senior Member

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    Godel's last theorem? Jesus.. this reminds me of Fermat's Last Theorem. If the two are alike in appeal, there will be a mad race to see who will be the first to produce a proof.
     
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  5. Pete It's not rocket surgery Registered Senior Member

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    I'm thinking Politics, Free thoughts, or maybe Science and Society. Which would you prefer?
     
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  7. Absane Rocket Surgeon Valued Senior Member

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    Free Thoughts may be a bad idea. I'd go Politics.
     
  8. §outh§tar is feeling caustic Registered Senior Member

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    I wouldn't say one's worse (than the other).

    Moreover, given the current leadership, I wouldn't worry too much about our esteemed leaders manipulating non-sequiturs in the Constitution to achieve villainous ends.

    Why, I think that's the least of our concerns for them.
     
  9. Prince_James Plutarch (Mickey's Dog) Registered Senior Member

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    Godel died from self-starvation because he was afraid that he would be poisoned by himself.

    Let's just say that the fucker wasn't right in the head.
     
  10. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    Yeah but he was brilliant.

    He was famous for his proofs by contradiction. For example, when Einstein claimed that GR contained only physically realizable solutions, Godel went and found a solution which contained closed time-like curves, effectively showing that GR allowed (at least in principle) time travel.

    His most famous theorem is something along the lines of "Every system of logically consistent arithmatic operations contains a truth which can't be proved." In other words, one can always find something which cannot be proved but must nontheless be true. Or math cannot be both consistent and complete. (Correct me if I'm wrong.)

    So this sort of caveat (i.e. the Constitution containing an important exception) is just the thing for Goedel.
     
  11. Absane Rocket Surgeon Valued Senior Member

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    From what I remember, this is correct. "Back in the day," some mathematician said we need to make mathematics complete because we want no holes whatsoever. As it turns out, this would mean we need to prove the axioms and as Godel showed, we cannot do this. Essentially, one cannot prove everything within the same system.

    Interestingly, if Fermat's Last Theorem fell into this catagory, this would imply that it is true. Why? Because if Fermat's Last Theorem were in fact false, then there would be a counter-example (integer solution) and this would be proof. Hence, FLT is true but there would be no proof that it is true. However, a proof did come about in *I think* 1994 so FLT does not fall into the category of statements that cannot be proved.

    Heh.. I love to prove things by taking the inverse of a statement then proving that by contradiction. I do this a lot with "if and only if" statements.




    But back to the topic. I'd love to see any research into this matter.
     
  12. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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    That's pretty much it, although there is an important qualifier that it only applies to sufficiently complex mathematical systems. Basically, they need to be sophisticated enough to encode statements about the validity of their own statements (i.e., "this statement is false."). There are plenty of useful mathematical theories to which the Incompleteness Theorem doesn't apply; Euclidean geometry, for example, is both complete and consistent. It's also possible to get around the theorem by specificying a theory in such a way that there is no algorithm for checking the validity of statements; you simply have an infinitely long list of axioms which exhaustively cover every possible true statement (this, of couse, is not a terribly elegant or even useful way to do math).

    This work was done in response to Hilbert's Program, where the idea was to prove the consistency of complex mathematical systems using simpler ones, and thus ultimately reduce the problem of proving the consistency of all mathematics to the problem of proving the consistency of arithmetic. Godel showed that you can't do this. However, the converse isn't true: you can indeed establish the consistency of simple systems using more complex ones. This still leaves you with the problem of establishing the consistency of the more complex systems, of course.
     
  13. Pete It's not rocket surgery Registered Senior Member

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    By all accounts, Kurt Godel was a most interesting guy. Genius mathemetician. Mad as a hatter. Conspiracy theorist. Died of starvation when his wife became ill and could no longer cook for him. Interestingly, Godel also produced an Ontological proof of God's existence.
     
  14. Farsight

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    Yes, I've read this. It was in A World Without Time: the forgotten legacy of Godel and Einstein by Palle Yourgrau. It's heavy on the philosophy, but it also contains some interesting physics history. For example, Godel's rotating universe was intended by him to disprove the possibility of time travel, not permit it.

    http://www.friesian.com/goedel.htm
     
    Last edited: Mar 28, 2007
  15. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    your example is clearly in a class of "self referencing" statements, and my favorate of the class. It is upon these that I pin most of my hope for a "logic" that permits free will to be consistent with physics. - For my strange position on this subject, see:

    http://www.sciforums.com/showpost.php?p=1294496&postcount=52

    This self referencing class of your example is not the same class as:

    "...sufficiently complex mathematical systems. ...sophisticated enough to encode statements about the validity of their own statements. ..."

    but is a sub group of this larger class.

    Can you give an example that is not a self referencing statement?

    I have always been confused about Godel's law (never had a course in logic - what little I know is self taught). For example, the four operations of ordinary math (defined on the real number line) contain the division of both 1 by 0 and 2 by 0. Yet one can prove that the second "infinity" is of the same size (not twice as large) as the first.

    Is this and example of Godel's rule applied to something as simple as four operations defined on the real number line? If it is not, because a special rule has been added for the "zero divide" case, is this not possible (give special rules added to the system) for all problems?

    Your comments would be appreciated. I know little about all this.
     
    Last edited by a moderator: Mar 28, 2007

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