Thanks, interesting reading.
Fundamental or not I think my point still stands that “the prior is infinite so the whole thing’s wrong” isn’t quite enough of an argument, since you still seem to conclude that improper priors can be used if used carefully enough. A more satisfying argument would be to demonstrate that the 9⁄10 case can’t be made without incorrect use of an improper prior. Though I guess it’s still showing where the problem most likely is which is helpful.
As far as being part of the foundations goes, I was just going by the fact that it’s in Jaynes, but you clearly know a lot more about this topic than I do. I would be interested to know your answer to the following questions though: “Can a state of ignorance be described without the use of improper priors (or something mathematically equivalent)?”, and “Can Bayesian probability be used as the foundation of rational thought without describing states of ignorance?”.
On the Doomsday argument, I would only take the Dice Room as a metaphor not a proof of anything, but it does help me realise a couple of things. One is that the setup you describe of a potentially endlessly exponentially growing population is not a reasonable model of reality (irrespective of the parameters themselves). The growth has to stop, or at least converge, at some point, even without a catastrophe.
It’s interesting that the answer changes if he rolls the dice first. I think ultimately the different answers to the Dice Room correspond to different ways of handling the infinite population correctly—i.e. taking limits of finite populations. For any finite population there needs to be an answer to “what does he do if he doesn’t roll snake-eyes in time?” and different choices, for all that you might expect them to disappear in the limit, lead to different answers.
If the dice having already being rolled is the best analogy for the Doomsday argument then it’s making quite particular statements about causality and free will.
I think this shows how the whole “language independent up to a constant” thing is basically just a massive cop-out. It’s very clever for demonstrating that complexity is a real, definable thing, with properties which at least transcend representation in the infinite limit. But as you show it’s useless for doing anything practical.
My personal view is that there’s a true universal measure of complexity which AIXI ought to be using, and which wouldn’t have these problems. It may well be unknowable, but AIXI is intractable anyway so what’s the difference? In my opinion, this complexity measure could give a real, numeric answer to seemingly stupid questions like “You see a number. How likely is it that the number is 1 (given no other information)?”. Or it could tell us that 16 is actually less complex than, say, 13. I mean really, it’s 2^2^2, spurning even a need for brackets. I’m almost certain it would show up in real life more often than 13, and yet who can even show me a non-contrived language or machine in which it’s simpler?
Incidentally, they “hell” scenario you describe isn’t as unlikely as it at first sounds. I remember an article here a while back lamenting the fact that left unmonitored AIXI could easily kill itself with exploration, the result of which would have a very similar reward profile to what you describe as “hell”. It seems like it’s both too cautious and not cautious enough in even just this one scenario.