User talk:Likebox/Gödel modern proof

(Another exposition of first paragraph which is less non-standard.) We may take our computer to be a Turing Machine without loss of generality. Any proposition of the form, "This Turing Machine will reach State X" can be expressed as a proposition in standard number theory.

(Then we pick up below at 'Such an axiom system is either inconsistent or incomplete.')

Postscript : Can we get to incompleteness even more directly? I'm not sure if this is correct (must think on it), but the proposition P-Halts is, using a Turing Machine, a proposition in number theory, call it NPH. If a proof exists for either NPH or its negation, then the P-Halts proposition is decided. Thus no proof exists for the number theoretic proposition NPH. CeilingCrash 06:14, 8 November 2007 (UTC)

You are right that this is equivalent, and in fact this is Carl's rewrite. The problem is that using a Turing machine or any other stupidity from the formal recursion theory literature is exactly what I am trying to avoid in this proof.Likebox 19:43, 8 November 2007 (UTC)

I didn't mean to insult you there. I sometimes sound like I am attacking a person when I am attacking an idea that I don't like for purely aesthetic reasons. I don't mean to.Likebox 06:01, 9 November 2007 (UTC)