How do i put Goedels theorem into a simple formula?

Discussion in 'Physics & Math' started by FrankExchangeOfViews, Apr 8, 2002.

  1. FrankExchangeOfViews Registered Member

    Hello everybody,

    i recently fell across the highly interesting matters of goedels incompletness theorem. Since i dont really have any algebraic or mathematical background, i need to ask for help: how can i put goedels theorem into a short formula? The purpose may sound confusing, but i need it for a t-shirt motive i want to create.

    I found this example by Rucker suitable for my level of undersanding to follow matters:

    << The proof of Goedel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:

    1. Someone introduces Goedel to a UTM, a machine that is supposed
    to be a Universal Truth Machine, capable of correctly answering any question at all.

    2. Goedel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.

    3. Smiling a little, Goedel writes out the following sentence:
    "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Goedel. Note that G is equivalent to: "UTM will never say G is true."

    4. Now Goedel laughs his high laugh and asks UTM whether G is true or not.

    5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.

    6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").

    7. "I know a truth that UTM can never utter," Goedel says. "I know that G is true. UTM is not truly universal."

    How can i put this into a short, striking formula? Does it sum up to something like (s for statment):

    s: s is false.

    Assuming this is summing it up, is there a way, for sole purposes of mystification, to replace ":" and "is false" with aequivalent mathemathical symbols? I know how to replace "is" with "=", but what aequivalent mathemathical symbold do i need to apply for ":" and "false"?



    Sorry for bad english

    -- "If you think that big government interferes in your life too much NOW, just wait till the government starts regulating the genetic constitution of your children" Theodore Kaczynski', "Industrial Society And Its Future"
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  3. nanok Registered Senior Member

    why would the machine even say that the sentence is true in the first place, godel seems to be TELLING the machine to never say that the sentence is true, why couldn't the machine just say that it's false???
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  5. FrankExchangeOfViews Registered Member

    If i follow the matter correctly....

    ... it comes down to this:

    lets say i pose you the statement "this is not true", then if it is not true, its actually true, but since i statet that it is not true, then its false. but when its false, then its true.... and so on. You will never be able to make a definitive conclusion.

    The philosophically interesting point is, that there is a place, where we cannot see. There is a point, where all knowlegde fails because there are questions, that cant be answered with certainity. This applies as well to the physical universe, where this manifests in black holes, a point where the laws of this universe fail.



    "Some of the symptoms listed are similar to those shown by caged animals"
    Theodore Kaczynski', "Industrial Society And Its Future"
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  7. oldesanc Registered Member

    Equivalent statements in concise forms:

    A system cannot prove itself by using explanations within itself.

    A system in thermal equilibrium cannot do output work by making itself as the heat source and the heat sink.

    Nobody can lift himself by pulling on his shoestrings.

    There's is no preferred frame of reference.

    No energy can be created within a system.

    In all the subsequent statements after the first, an outside explanation is needed if the opposite occurs.

    In short Goedel's theorem puts a limitation on the explanatory power of science and mathematics.
  8. CANGAS Registered Senior Member

    Everything I have ever told you, am telling you now, or will tell you, is a lie. Really.

    Just kidding.
  9. funkstar ratsknuf Valued Senior Member

    Hmm. A formula is going to be difficult to provide, because it will depend very much on the notation. A sentence describing the essense might be better. Something like "There are true statements that cannot be proved."

    (Here's a little teaser to test your understanding of the thought behind Gödel's theorem: Why, in the sentence above, does the "true statements" not need the qualifier "about numbers" (or "number theory")?)

    If you still want a symbolic representation, the following is quite good:

    &exist; x.

    Please Register or Log in to view the hidden image!

    x &and;

    Please Register or Log in to view the hidden image!


    The "|="* is a standard symbol for semantic entailment, so "|= x" means that x is true. Similarly, "|-" is a standard symbols for provability, so "|-" x means that x is provable. The dash across the symbol is a standard way to write negation. Hence, the statement above reads "there exists x such that x is true and x is not provable."

    *I have to use this representation, because the board is telling me I have to many images, otherwise. Oh, to have LaTeX...

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