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ilyasut's profile
Ilya Sutskever
Ilya Sutskever
Ilya Sutskever
@ilyasut

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Ilya Sutskever

@ilyasut

research @openai

Joined September 2013

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    Ilya Sutskever‏ @ilyasut 3 May 2019

    A short proof of Godel's thm: assume every statement or its negation has a proof. Consider the statement "Turing machine M halts". Keep enumerating all proofs until we find a proof that M halts or a proof of its negation. Thus we've solved the halting problem, contradiction!

    11:01 AM - 3 May 2019
    • 24 Retweets
    • 167 Likes
    • Aravind Srinivas Massimo Guadagno barton 🥕 Seungjae Ryan Lee Renaud Bougueng βότσαλα CollaboRational Jason Mancuso Giorgio Patrini
    10 replies 24 retweets 167 likes
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      1. Polatbilek‏ @ozanpolatbilek 3 May 2019
        Replying to @ilyasut

        Actually Turing used Gödel’s approach to prove halting problem is unsolvable within Turing Machines. So using Halting Problem to prove Gödel’s incompleteness theorem looks like a recursive loop. Halting problem proves Gödel, gödel proves halting problem.... goes on.

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      2. Christian Szegedy‏ @ChrSzegedy 3 May 2019
        Replying to @ilyasut

        This is essentialy the essence of the original proof, the technical part is that the Peano exioms can encode this statement on Turing machines.

        1 reply 0 retweets 7 likes
      3. Sven Cattell‏ @comathematician 3 May 2019
        Replying to @ChrSzegedy @ilyasut

        I love that the first proof boils down to a very clever extension of the "this statement is false" trick we teach kids.

        1 reply 0 retweets 1 like
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      2. Volodymyr Kuleshov‏ @volokuleshov 3 May 2019
        Replying to @ilyasut

        It's also interesting that proofs of both Godel's theorem and the halting problem are based on a similar diagonalization argument, which also suggests that they are closely related.

        2 replies 0 retweets 6 likes
      3. Shubhendu Trivedi‏ @_onionesque 3 May 2019
        Replying to @volokuleshov @ilyasut

        Uncomputability is the deeper reason for incompleteness. For Turing showed something stronger i.e. soundness + completeness implied you could systematically settle anything you query your formal axiomatic system.

        1 reply 0 retweets 2 likes
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      1. David T.‏ @david_networks 4 May 2019
        Replying to @ilyasut

        You mean Refutational Reasoning?

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      2. François Fleuret‏ @francoisfleuret 4 May 2019
        Replying to @ilyasut

        The puzzling part of Gödel theorem IMO is that there are undecidable propositions of the form "exists k, s.t. P(k)" since the existence of such a k would provide a proof, so k cannot exist, so I do not understand "exists k".

        1 reply 0 retweets 1 like
      3. Alexander Clark‏ @alexandersclark 4 May 2019
        Replying to @francoisfleuret @ilyasut

        The k is not a natural number but an element of a non standard model of arithmetic which does not code for a proof.

        1 reply 0 retweets 0 likes
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