On 14 Aug 2009, at 03:21, Colin Hales wrote:

> Here's a nice pic to use in discussion.... from GEB. The map for a  
> formal system (a tree). A formal system could not draw this picture.  
> It is entirely and only ever 'a tree'. Humans dance in the forest.
> col



You may compare Hofstadter's picture with the Mandelbrot set, and  
understand better why it is natural to think that the Mandelbrot set  
(or its intersection with Q^2) to be a "creative set" in the sense of  
Emil Post, that is, mainly, a (Turing) Universal system. The UD* (the  
block comp multiverse) can be mapped in a similar way.

See here for a picture of the Mnadebrot set (and a comparison with  
Verhulst bifurcation in the theory of chaos):

http://en.wikipedia.org/wiki/File:Verhulst-Mandelbrot-Bifurcation.jpg

Or see here for a continuos enlargement:

http://www.youtube.com/watch?v=RTuP02b_a7Y&feature=channel_page

Or perhaps better, in this context, a black and white enlargment:

http://www.youtube.com/watch?v=UrEoKFYk0Cs&feature=channel_page

or a 3-d version

http://www.youtube.com/watch?v=zciBjiD9Zfg&feature=channel_page

Colors or eights help to see the border of the set, but it is really a  
subset of R^2. The border is infinitely complex, but not fuzzy! It is  
really a function from R^2 to {0, 1}.

Bruno


http://iridia.ulb.ac.be/~marchal/




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