Re: Trust, Consciousness, Dennett, Lem: was: another new language to check out

Sally Caves wrote:

>Hi, Chris.  I agree with anything you say here, and welcome your added
>remarks!  :)  I've heard of Go:del's Incompleteness Theory, I've heard of
>String Theory, the point is I've HEARD of all these theories, but I haven't
>formally studied them, only taken out a few books for the layman.
>
I can help you if you want.  An excellent summary of Goedel's
Incompleteness Theorem is found in Robert Penrose's book 'The Emperor's
New Mind' - which, essentially, covers everything important about
Mathematics at once, in order to come to some theory about the mind.  I
haven't got to that bit yet, it's rather hard going.

I can try to give an idea of it here, if you like.

Imagine, for a moment, an alphabetic ordering of every single
Mathematical proposition possible(based on a limited number of axioms -
ideas that can be taken for granted within a system). So, P1(w) is the
first proposition, applied to the number w.  Now, let us take another,
similarly alphabetic system, with all proofs of various propositions
arranged.  Qx is the xth proof, Pn is the nth proof

Now, consider the statement 'There is no value of x, such that Qx proves
Pw(w)".  That is, Pw(w) cannot be proven.  Now, as every proposition has
a number in the list of P, we can assign the previous proposition a
number.  Let us say,  k.  Thus,  Pk(w)  means the previous statement.
Now, if we give w the value of k, Pk(k) means 'There is no value of x,
such that Qx proves Pk(k)'. Assume Pk(k) is false.  That is, it can be
proven.  Evidently this is absurd.  A false statement cannot be proven.
Thus, Pk(k) cannot be proven, as assuming it is true also leads to that
conclusion.  Therefore, Pk(k) is true, but cannot be proven *within its
own system*.