1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
2. Gödel 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, Gödel 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 Gödel. Note that G is equivalent to: “UTM will never say G is true.”
4. Now Gödel 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,” Gödel says. “I know that G is true. UTM is not truly universal.”
(via)



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7 Responses

  1. delio says:

    non so. non sono un logico, ma a me sembra che quello presentato sia molto più una variante del paradosso del cretese che una spiegazione del teorema di incompletezza di goedel (che non ha caso ha bisogno di circa un centinaio di pagine nella sua forma divulgativa più semplice): “in ogni sistema formale che contiene la teoria aritmetica elementare è possibile costruire un asserto aritmetico che sia vero ma né dimostrabile né falsificabile, ammesso che la teoria sia consistente”.

    27 July 2005 @ 09:15

  2. Giavasan says:

    Sono d’accordo con te. Ancora non sono sicuro di aver capito bene il teorema, nonostante abbia letto più libri al riguardo. Semplicemente mi divertiva l’idea della versione “for dummies”.

    27 July 2005 @ 13:45

  3. proctorul says:

    this is picturesque and helpful.

    28 January 2010 @ 07:02

  4. Alta says:

    So what is the truth, if Maths and logicial thinking can’t be true? Isn’t it time for us to start thinking in the opposite direction? Einsten said that idiocy is when you keep on doing the same mistake over and over, and don’t realise that it is useless. What (or whom) is the truth?? Even though the Truth stood in front of him, Pilate couldn’t recognize it….

    14 February 2010 @ 09:45

  5. Gödel Summarized « ricketyclick says:

    […] Here’s your Sunday Morning Brain Stretcher: Gödel’s Incompleteness Theorem for Dummies. […]

    21 March 2010 @ 17:33

  6. r says:

    this is russel’s paradox and has nothing to do with godel?

    3 July 2011 @ 08:42

  7. ramgorur says:

    Well,

    “If God is omnipotent, then could He create a large rock that He Himself will not be able to lift it” ?

    something like it?

    10 May 2012 @ 07:06