"Making harder the conundrum is that Godel's Incompleteness and Turing's halting only apply to countable sets, not continua."
-- Precisely, but the riddle being that an infinitely countable set is included in that definition...and it's possible to have infinite subsets of such sets! So discretization (of anything) seems to be fundamental to continuity...this truth of mathematics which stands apart from our present understanding of reality may point us in the direction of what reality really is about.
If the pattern continues to apply it may provide a way to test validity of multiverse theories. As you may know, there are several interpretations of the results of quantum mechanical theory and general relativity that seem to indicate the flowering of either event histories, or the multiplicity of entire Universes...but the question is, are these histories or Universes real? Which is the truth? Mathematically, the application of sum over histories in Feynman's formulation of QED enabled the solving of previously intractable problems. The solution involved the idea of wave function collapse (sum over) over an *infinite* set of possible histories for a given event..each with variable probability of occurrence.
Some interpreted wave function collapse as meaning the spontaneous creation of new realities to evolve the state of the system, others see the multiverse as a coeventful (I can't say "temporal" since measurement of time may be different between Universes, so I use "event" to encapsulate the relative difference in time measurement that would attend different Universes...according to GR, having a time dimension at all is not even necessary) set of Universes that may or may not exist *now* but did at some time in the past or will exist at some time in the future.
I attended a Physics symposium in 2010, where I was fortunate to meet and hear speak David Gross, Micio Kaku and Alan Guth. During a spirited discussion Gross and Guth battled over whether or not they felt the idea of multiverse made sense. Guth imagined there could be a multiverse of universes and posited possible travel between them...Gross called it nonsense saying that the mathematical existence of multiple Universes does not guarantee coeventful transit capability (which I agree with).
I am reminded of how we now understand energy in the modern sense that is different from how it was described classically when one studied 19th century energy based on newtonian physics. Then, we were told that "energy can't be created or destroyed" in the famous second law, it turns out quantum allows it to be created and destroyed so long as over time those events even out. *There is a sum over the histories* of not just particle evolutions but particle existence...if that's the case it means a) all of this truly can be the result of the spontaneous eruption of something from nothing. and b) that in order for balance to be restored this "something" will and must go back to a "nothing" state. This though sounds a lot like what was supposed earlier about a coeventful multiverse. What if it is enough that the ensemble (infinite) of Universes goes through all possibilities for each contained particle event... *eventually* rather than that they all occur coeventfully. It would solve the multiverse problem by applying Feynman's idea of sum over histories to the entire multiverse itself. It's actually beautiful....and fractal at core...and also...in it's universal absolute...very binary, like the divisions between the integers upon which both Godels incompleteness and Turing's computation halting problem are based....coincidence?? I don't think so.
""What if" recursion (iow fractal nature) is so Total, it's wholeclothe with dimensionality?"
I considered this but it would seem from what we know thus far that reality is not "turtles all the way down" as we look deeper into the fabric we see discretization not continuity. We go down to the atom and see it begin in energy transitions of electrons in their "orbits"....allowed and forbidden with no middle ground, we see the emergence of macroscopically absent dimensions of discretized energy transduction...like "spin" (intrinsic angular momentum) ....we go down further still and see discretization in the family of quarks that make up the particles of the atom. Deeper still and we reach the limit of the planck length...below which space itself is a roiling soup of undulations that emerge and eliminate particles. For the recursion idea to hold *within our universe* there would have to be a continuous ability to probe deeper...so far all our physics...theoretically falsified or unfalsifiable (string theories) point to a bottom. A possible solution though lays in if additional universes *pinch off* dimensionally from parent Universes such that all their dimensions are orthogonal to all ours...this would then make them independent and maintain the fractal symmetry but instead of going "down" it goes sideways...as it were. This idea is one I am partial to (and one that received media last year when a physicist proposed it)...and the fact that GR predicts objects that are forever hidden from the Universe that created them (black holes...namely the singularity within) it just may be the "sideways" set of dimensions in which new Universes (and their physics and ability to create black holes) exist....and thus maintain "turtles all the way sideways". The continuity is preserved through the fractal recursion...the discretization is preserved through the independence of the new Universes birthed...riddle solved.