"...Gödel statements must be true, even though they can't be proved algortihmically."
It depends on what you mean by 'algorithmically'. Any set of logical operations to input data can be translated as an algorithm. Same thing regarding logical tests and philosophical arguments like 'reductio ad absurdum'. What is exactly the part of the process a machine can't perform?.
I can understand K.Gödel using such a limited definition for algorithm, as IT was an emerging field back then, but this argument seems to have aged poorly.