Perhaps you've confused the "pi has a decimal expansion goes on forever without seeming pattern" with "a generic real number has a decimal expansion that goes on forever without pattern"? Pi does have a finite representation, "Pi". We use it all the time, and it's just as precise as "2" or "0.5".
Specifically, you could start with the rationals, and complete it by including all solutions to differential equations. Pi would be included, and many other numbers, but you'd still only have a countable set - because every number would have one or more shortest definitions - finite information content.
If you had a probability distribution over such a set, it would naturally favor hypotheses with short definitions. If it started out including pi as a possibility, and you gathered sufficient sensor data consistent with pi (a finite amount), the probability distribution would give pi as the best hypothesis. This is reasonable behavior. You have to do non-obvious mucking around with your prior to get that sort of reasonableness with standard real-number probabilities.
As others have pointed out, any specific countable system of numbers (such as the "solutions to differential equations" that I mentioned) is susceptible to diagonalization - but I see no reason to "complete" the process, as if being susceptible to diagonalization were a terrible flaw above all others. All the entities that you're actually manipulating (finite precision data and finite, stateable hypotheses like "exactly 2" and "exactly pi") are finite-information-content, and "completing" the reals against diagonalization makes essentially all the reals infinite-information-content - a cure in my mind far worse than the disease.
(Note: I’m not arguing in this particular post, just asking clarifying questions, as you seem to have the issues much clearer in your mind than I do.)
1) It seems one can start with naturals, extend them to integers, then to rationals, then to whatever set results from including solutions to differential equations (does that have a standard name?). I imagine there are countably infinite many constructions like that, am I right? They seem to “divide” the numbers “finer” (I’d welcome a hint to more formal description of this), though they aren’t necessarily t...
Today at lunch I was discussing interesting facets of second-order logic, such as the (known) fact that first-order logic cannot, in general, distinguish finite models from infinite models. The conversation branched out, as such things do, to why you would want a cognitive agent to think about finite numbers that were unboundedly large, as opposed to boundedly large.
So I observed that:
And the one said, "Isn't that a form of Pascal's Wager?"
I'm going to call this the Pascal's Wager Fallacy Fallacy.
You see it all the time in discussion of cryonics. The one says, "If cryonics works, then the payoff could be, say, at least a thousand additional years of life." And the other one says, "Isn't that a form of Pascal's Wager?"
The original problem with Pascal's Wager is not that the purported payoff is large. This is not where the flaw in the reasoning comes from. That is not the problematic step. The problem with Pascal's original Wager is that the probability is exponentially tiny (in the complexity of the Christian God) and that equally large tiny probabilities offer opposite payoffs for the same action (the Muslim God will damn you for believing in the Christian God).
However, what we have here is the term "Pascal's Wager" being applied solely because the payoff being considered is large - the reasoning being perceptually recognized as an instance of "the Pascal's Wager fallacy" as soon as someone mentions a big payoff - without any attention being given to whether the probabilities are in fact small or whether counterbalancing anti-payoffs exist.
And then, once the reasoning is perceptually recognized as an instance of "the Pascal's Wager fallacy", the other characteristics of the fallacy are automatically inferred: they assume that the probability is tiny and that the scenario has no specific support apart from the payoff.
But infinite physics and cryonics are both possibilities that, leaving their payoffs entirely aside, get significant chunks of probability mass purely on merit.
Yet instead we have reasoning that runs like this:
(Posted here instead of Less Wrong, at least for now, because of the Hanson/Cowen debate on cryonics.)
Further details:
Pascal's Wager is actually a serious problem for those of us who want to use Kolmogorov complexity as an Occam prior, because the size of even the finite computations blows up much faster than their probability diminishes (see here).
See Bostrom on infinite ethics for how much worse things get if you allow non-halting Turing machines.
In our current model of physics, time is infinite, and so the collection of real things is infinite. Each time state has a successor state, and there's no particular assertion that time returns to the starting point. Considering time's continuity just makes it worse - now we have an uncountable set of real things!
But current physics also says that any finite amount of matter can only do a finite amount of computation, and the universe is expanding too fast for us to collect an infinite amount of matter. We cannot, on the face of things, expect to think an unboundedly long sequence of thoughts.
The laws of physics cannot be easily modified to permit immortality: lightspeed limits and an expanding universe and holographic limits on quantum entanglement and so on all make it inconvenient to say the least.
On the other hand, many computationally simple laws of physics, like the laws of Conway's Life, permit indefinitely running Turing machines to be encoded. So we can't say that it requires a complex miracle for us to confront the prospect of unboundedly long-lived, unboundedly large civilizations. Just there being a lot more to discover about physics - say, one more discovery of the size of quantum mechanics or Special Relativity - might be enough to knock (our model of) physics out of the region that corresponds to "You can only run boundedly large Turing machines".
So while we have no particular reason to expect physics to allow unbounded computation, it's not a small, special, unjustifiably singled-out possibility like the Christian God; it's a large region of what various possible physical laws will allow.
And cryonics, of course, is the default extrapolation from known neuroscience: if memories are stored the way we now think, and cryonics organizations are not disturbed by any particular catastrophe, and technology goes on advancing toward the physical limits, then it is possible to revive a cryonics patient (and yes you are the same person). There are negative possibilities (woken up in dystopia and not allowed to die) but they are exotic, not having equal probability weight to counterbalance the positive possibilities.