User blog comment:KthulhuHimself/The Mandelbrot function, and the use of fractals in googology./@comment-1605058-20151018135434

I am not entirely sure if this function is everywhere well-defined (the boundaries of sets considered might not have their curvature well-defined, or there might be infinitely many ), but here is a sketch of a proof that if it is well-defined, then it's computable:

First, we have to note that complex numbers can be interpreted as pairs of real numbers, and standard arithmetical operations on them are first-order definable (in language of theory of ). Secondly, for fixed n, the function \(P_c^n(0)\) is a polynomial in \(c\), and \(c\) is in the "n+1 iterated" set iff \(|P_c^n(0)|\leq 2\). We then can use this to first-order define points which lie on the boundary of this set. Thirdly, we have to define curvature and extreme points of curvature. I will handwave over this by saying that using standard epsilon-delta definitions of limits, derivatives and extremes will work.

Hence there is a first-order formula \(E(x,y)\) which means that \(x+yi\) is the extreme point of curvature for nth iterate set. To compute Mb(n), we will use the fact that : the algorithm decide the truth of the following formulas, from \(k=1\): "there exist numbers \(x_1,y_1,...,x_k,y_k\) such that \((x_i,y_i)\) form \(k\) distinct pairs and \(E(x_i,y_i)\) for \(i\) from 1 to \(k\)". If \(k+1\) is the first for which this formula is false, then Mb(n)=k.