On 23 Jan 2014, at 06:03, Russell Standish wrote:
On Wed, Jan 22, 2014 at 11:12:50AM +0100, Bruno Marchal wrote:
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.
Yes - I recall that was how the Wikipedia article defined it. But I
don't grok it. What is the motivation for such a definition? What
about some examples (I'm guess the Mandelbrot set might be one such)?
Well, with Myhill's theorem showing that Turing complete set =
creative set, you can generate a creative set with any universal
machine, or universal programming language.
Take its enumeration phi_i.
The set of x such that phi_x(x) converge (is defined) is creative.
The set of <x,y> such that phi_x(y) converge is creative.
Like I just said to Brent, the set of provable statement in any RE
extension of RA (and thus of PA, of ZF, of NF, ...) is creative.
If comp is true, we can say that your local (here and now) accessible
rational (with my definition) beliefs (accessible in principle in
platonia) is creative, so *you* are an example, but your laptop is too.
The John Myhill proved that a set is creative iff it is Turing
complete, i.e. Turing universal.
So that RE set
What is a Turing complete _set_?
It is a set formal view of a universal Turing machine. When doing
computability theory, we can decide to concentrate on the w_i, instead
of the phi_i. (with w_i = the domain of the phi_i).
Bruno
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Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics [email protected]
University of New South Wales http://www.hpcoders.com.au
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