Dear Markus,
thanks for reading my essay, and for commenting!
You are quite right in your observation that my argument, basically, is just equal to Cantor's regarding the fact that the powerset of a set necessarily has a greater cardinality than the set itself, and hence, that there can be no bijection between the two. This is a very familiar fact to us, today, but still, depending on the context, has quite nontrivial implications---the fact that there are uncomputable functions, or indeed, undecidable statements in any sufficiently expressive theories, follow exactly the same fold.
It's not quite right to say that this applies equally well to a quantum world, however. The reason for this is that the basic underlying structure---Lawvere's fixed-point theorem---works in the setting of Cartesian closed categories; the category Hilb that has Hilbert spaces as its objects, and linear operators as its morphisms is not Cartesian closed, however. Baez has provided an interesting discussion on how it's exactly this categorical difference that underlies most of the 'weirdness' of quantum theory.
In particular, the absence of a cloning operation means that the diagonalization doesn't go through---you can't, in a sense, feed the system back the information about the system. So in that sense, my argument entails that sets aren't a good setting for a physical theory, as you run into the paradoxical, and you have to adduce extra structure (by a deformation of the algebra of observables) to avoid this---which leads to something like phase-space quantization. Or, alternatively, you can start out with a categorical setting where you get this structure for free---leading to something like Hilb.
Bohmian mechanics, by the way, isn't a counterexample---indeed, I think it supports my argumentation (this is discussed more in depth in the Foundations of Physics-article). In the end, it comes down to the fact that every function---including noncomputable ones---can be represented by means of a finite algorithm, augmented with an infinite string of random digits (every set is reducible to a random set by the Kucera-Gacs theorem). In general, thus, every measurement outcome in Bohmian mechanics is a function of the entire random initial conditions---which must fit the equilibrium hypothesis to give rise to quantum predictions. (Indeed, if the generation of measurement outcomes in Bohmian mechanics were computable, that would lead to exploitable nonlocal signalling.)
Indeed, that's to me at least a suggestive way of forming the connection to quantum mechanics: a noncomputable function (or sequences) can be 'computed' in different ways---one, with a finite algorithm with interspersed random events, two, with a finite algorithm that reads out a fixed algorithmically random number, three, with an interleaving process computing every possible sequence. These correspond to the major interpretations of quantum mechanics---something like a Copenhagen collapse process, with the algorithm being the Schrödinger dynamics (von Neumann's 'process II'), and random events yielding the 'collapse' ('process I'), a Bohmian nonlocal hidden-variable approach, and a kind of many worlds theory.
That said, I view this as very much a sketch of a theory---perhaps itself a kind of toy theory. To me, it seems a promising avenue to investigate, but I have no illusions about having painted any sort of complete picture at all. I ride roughshod over many subtleties, as you note; and there are several additional open questions. Some of this is treated more carefully in the Foundations of Physics-paper (which also properly cites the work by Hoehn and Wever---well, not quite properly, since I call him Höhn!), where I am also more cautious about some of my claims. There, also an argument based on Chaitin's incompleteness theorem, that doesn't boil down to 'mere diagonalization', is included.
Thanks, again, for taking the time to read and comment on my essay. I would very much enjoy continuing this discussion---since I work on this mostly in isolation, there's a high danger of getting lost down blind alleys, so I welcome any reality check on my ideas. So any and all criticism is greatly appreciated!
Cheers
Jochen