Off the top of my head, it sounds like you're going to run into computability problems.

I've annoyingly forgotten the terminology and don't have time to look it up, and some details may be incorrect, but: some algorithms can be written in a programming language which imposes bounds on their running time before the algorithm is actually run. (The runtime is allowed to depend on the arguments to a function.) Others can only be written in a language which allows you to write infinite loops. (This is true even of some algorithms which are guaranteed to terminate on all inputs.)

The algorithms you iterate over will have to be of the first kind, or you'll quickly run into an infinite loop. But the algorithm to do the iteration and diagonalise will be of the second kind. Thus, no paradox.

(IIRC, exactly this problem was discussed in GEB.)