But then I had a whimsical thought; the sort of thing that seems at once not-impossible and yet such a long shot that one can just relax and enjoy exploring it without feeling under pressure to produce a result in any particular timeframe (and yet, I have moved my thinking forward on this over the years, which keeps it interesting).

What if we could find a way to take advantage of the fact that our logic is embedded in a computational system, by somehow bleeding off the paradoxes into mere nontermination? So that they produce the anticlimax of functions that don’t terminate instead of the existential angst of inconsistent mathematical foundations ?