On 9/14/06, Nick Hay <[EMAIL PROTECTED]> wrote:
You're both right. Ben's quote is a famous one made my Chaitin,
see http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html
The problem is with how you define the "weight" of a theorem.
In this context "weight" means that the theorem itself contains a
large amount of algorithmic information, i.e. has a high Kolmogorov
complexity. Thus "x=x" is a very "light" theorem in this sense.
The key question is, what would a friendliness theorem be about?
Would it be about a function where the data that goes into and out of
it aren't important? For example, "x=x". Or would it be a theorem about
the complete system of the AGI *and* universe that it interacts with?
If it is the latter, then the theorem will have to be very heavy, unless
you can come up with a simple theory to explain the universe. As
Chaitin points out, theorems that are "heavy" in this sense cannot
be proven due to Gödel incompleteness.
ShaneOn 9/13/06, Ben Goertzel <[EMAIL PROTECTED]> wrote:
> The basic problem as many have noted is Godelian. Chaitin's version
> of Godel's Theorem says "You can't prove a 20 pound theorem with a 10
> pound axiom system." We humans cannot prove theorems about things
> that are massively more algorithmically complex than ourselves.
That's not true. A trivial example: very simple number theories can
prove "n = n" for any value of n, no matter how large. In particular
for extremely complex n.
More interestingly, we can prove some things about the behaviour of
sufficiently simple programs no matter how complex their input is.
You're both right. Ben's quote is a famous one made my Chaitin,
see http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html
The problem is with how you define the "weight" of a theorem.
In this context "weight" means that the theorem itself contains a
large amount of algorithmic information, i.e. has a high Kolmogorov
complexity. Thus "x=x" is a very "light" theorem in this sense.
The key question is, what would a friendliness theorem be about?
Would it be about a function where the data that goes into and out of
it aren't important? For example, "x=x". Or would it be a theorem about
the complete system of the AGI *and* universe that it interacts with?
If it is the latter, then the theorem will have to be very heavy, unless
you can come up with a simple theory to explain the universe. As
Chaitin points out, theorems that are "heavy" in this sense cannot
be proven due to Gödel incompleteness.
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