Dear Jochen,
I finally had time to read your essay. I really appreciated the clarity of your arguments.
Your first example of the derivation of Heisenberg's principle from Finiteness and Extensibility is enlightening. Your introduction to superposition from diagonalisation is also very interesting. However, as you pointed out in your essay, "quantum mechanics, itself, does not fall prey to the same issues", notably because of the no-cloning theorem and the fact that the diagonal state yields a fixed point for the X gate. The fact that the superposition allows to avoid a logical contradiction reminds me of escaping the self-referential paradoxes by invoking a many-valued logic (e.g. trivalent), where '' (or "indeterminate") would be another kind of truth value, in addition to 0 and 1. But what about complex states ? I don't know if you are familiar with it, I heard about a book entitled 'Laws of Form' by Spencer-Brown which presents a calculus dealing with self-reference without running into paradoxes, by introducing an imaginary Boolean algebra. Take the equation x=-1/x , which entails in some way a self-reference, a mimic of the Liar. If x=1 then it is equal to -1 and vice-versa, leading to a contradiction. The solution is to introduce an imaginary number, i , defined by i=-1/i.
Your reading of Bell's theorem as revealing a counterfactual undecidability was enjoyable to read, as it is in line with my presentation of contextuality as a similar undecidability.
Another point : As you may have read as an epilogue in my essay, I am also interested in the Liar like structure that can emerge from "physical (hypothetical) loops" like CTCs. I am especially interested in quantum-based simulations of such CTCs, as Bennett and Loyd's P-CTCs. In the literature, e.g. https://arxiv.org/abs/1511.05444, people have studied the relation between logical consistency and the existence of unique fixed-point. I was wondering if you had also some kind of epistemic reading of such loops, if you think that this is also related to Lamvere's theorem.
Cheers,
Hippolyte