[Legend2440]

We can give them both infinite precision, you still can't build a lookup table of the mandelbrot set.

The mandelbrot set is essentially a map of the halting behavior of a specific program. You can't know whether or not the program will halt for a given input, and so cannot build the lookup table. Programs are stronger than input-output mappings.

[ben_w]

Infinity is really hard to reason about, are you sure about that?

(For all I know you're a PhD in transfinites, your profile says nothing).

[somewhereoutth]

Infinity (of the various kinds) is well understood (see Cantor etc).

The Halting Problem is a central result in computer science, again well understood (especially here I would think!)

Their comment is correct.