On 23 Jan 2014, at 01:15, meekerdb wrote:
On 1/22/2014 2:12 AM, Bruno Marchal wrote:
On 22 Jan 2014, at 01:41, Russell Standish wrote:
On Tue, Jan 21, 2014 at 12:53:33PM +0100, Bruno Marchal wrote:
With some competence, I guess you mean.
Without competence, and giving time to the creature, any universal
machine do have an open-ended creativity. Well, certainly in the
sense of Post (I can explain this, but it is a bit technical).
I'm interested to hear your explanation, but if its what I suspect
it
will be, I'll be disappointed :).
A set (of natural numbers) is creative if
1) it is RE (and thus is some w_k)
2) its complement (N - w_k) is productive, and this means that for
all w_y included in, we can recursively (mechanically) find an
element in it, not in W_y.
It means that the set is RE and his complement is constructively
NOT RE. Each attempt to recursively enumerate he complement can be
mechanically refuted by showing explicitlky a counterexample in it,
and this gives the ability to such a creative set to approximate
its complement in a transfinite progressions of approximation. this
gives an ability to jump to a bigger picture out of the cuurent
conception of the big picture. I find it a reasonable definition of
creativity.
So what would be an example of a creative set of natural numbers?
Take RA, or PA, or ZF. The set of the Gödel numbers of the provable
statements in any such theories is creative.
Are there sets of natural numbers such that both the set and its
complement are not RE?
Yes. Take a universal programming language.
Take the set of the gödel numbers of the code of the TOTAL functions.
It is not RE (we have proven this), nor is his complement RE.
Or take the set of the programs which computes the factorial
functions. The same thing will happen.
Or consider a theory like PA. And consider the set of true sentences
(in the standard model). It is not RE, nor his complement.
Bruno
Brent
The John Myhill proved that a set is creative iff it is Turing
complete, i.e. Turing universal.
So that RE set
Basically stating that the universal dovetailer emulates creative
conscious being does not demonstrate a creative program, which needs
to be creative relative to us (as observers).
I agree. The UD is not creative. But it generates all creative
programs or sets.
Note that the UD can be considered as creative though, if you
conceive it as the set of all initial segment of UD*. In particular
the set define by the diophantine polynomial that I send today to
Brent, *is* probably creative itself.
But if your idea is something different, I'm all ears!
I haven't had a chance to study and understand Post's definition
(sure
I've looked at it, but didn't grok it), but if you say it is
equivalent to universality, then its not really going to
contribute to
the solution.
I am not sure. Open ended creativity seems to me well captured by
Post. It makes the machine able to defeat all effective complete
theories about itself. It gives what I often called the comp
vaccine
against reductionism.
Well - maybe if you explain more?
I hope that what is above is not too much concise.
Bruno
Cheers
--
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Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics hpco...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
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