@paulkreinerhere @schulzb589 I accept that I haven't given a proof of this impossibility, but I believe the concerns I've laid out could probably be formalised in a way that would lead to one, though it'd involve something like a combinatoric diagonalisation similar to Russell's paradox.

@schulzb589 To state my issue in a more philosophical way, I think the desire to combinatorially totalise mathematical syntax is an effect of something like a transcendental illusion, what I sometimes call the Mythos of Logos.

@schulzb589 It's the Leibnizian desire to obviate creative dialogical interaction (dialectical reason) by providing a deductive system (monological reason) that would allow us to simply 'shut up and calculate'; the desire to obviate the pragmatic autonomy of discovery by automating it.

@schulzb589 It doesn't matter how many times this desire is betrayed by its own formal tools: Russell, Godel, Turing, Chaitin, etc.; it doesn't matter that these betrayals are what drive the externalisation of thought by producing new tools to automate computation. It always comes back.

@schulzb589 The desire to brute force reality, be it mathematical or empirical, is always lying in wait, whispering seemingly 'reasonable' lies about what is and isn't rationally possible. This is as close as I come to endorsing Heidegger and Adorno's worries about technoscience.