The Mandelbrot set is a beautiful creature in Mathematics. There are a lot of be

Question

The Mandelbrot set is a beautiful creature in Mathematics.

There are a lot of beautiful images of this set created with high precision, so obviously this set is "computable" in some sense.

However, what concerns me is the fact that it is not even recursively enumerable - simply because the set is uncountable. This could be resolved by requiring some sort of finite representation of the points.

Furthermore, although we know for sure that a lot of points belong to the set and others do not, there are also a lot of points whose membership in the set we don't know. All the images we've seen so far may include a lot of points that "up to n iterations kept in bound," but those points may not actually belong to the set.

So, for a given point with a finite presentation, the problem "Does this point belong to the set?" has not been proved to be decidable yet, if I am right.

Now, in what sense (by which definition) can we say the Mandelbrot set is "computable"?

Explanation / Answer

There are several ways of defining what it means for the Mandelbrot set to be computable. One possible definition is the Blum

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