I found a paper on the Mandelbrot set and computability, I understand very 
little but maybe Bruno would be able to follow it:

The same author has a shorter outline or slides for a presentation on this 
subject at 
 and at the end he asks the question "If M (Mandelbrot set) not Q-computable, 
can the Halting Problem be reduced to determining membership of (intersection 
of M and Q^2), i.e. how powerful a 'hypercomputer' is the Mandelbrot set?" I 
believe Q^2 here just refers to the set of all possible pairs of rational 
numbers. Maybe by "reducing" the Halting Problem he means that for any Turing 
machine + input, there might be some rule that would translate it into a pair 
of rational numbers such that the computation will halt iff the pair is 
included in the Mandelbrot set? Whatever he means, it sounds like he's saying 
it's an open question...
> On Thu, Apr 30, 2009 at 10:35 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:
>> The mathematical Universal Dovetailer, the splashed universal Turing
>> Machine, the rational Mandelbrot set, or any creative sets in the
>> sense of Emil Post, does all computations. Really all, with Church
>> thesis. This is a theorem in math. The rock? Show me just the 30 first
>> steps of a computation of square-root(2).       ...
> Bruno,
> I am interested about your statement regarding the Mandelbrot set
> implementing all computations, could you elaborate on this?
> Thank you,
> Jason
> > 

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