Although this is fighting the hypothetical, I think that the universe is almost certainly infinite because observers such as myself will be much more common in infinite than finite universes. Plus, as I’m sure you realize, the non-zero probability that the universe can support an infinite number of computations means that the expected number of computations we expect to be performed in our universe is infinite.
As Bostrom has written, if the universe is infinite then it might be that nothing we do matters so perhaps your argument is correct but with the wrong sign.
That doesn’t rule out infinite computation, though, since in an infinite universe we have a perpetually increasing amount of resources (as we explore further and further at lightspeed).
No, I was referring to inflationary space-time. The fact that the universe is still expanding (and accelerating in its expansion) means that 92% of the observable universe can never be reachable by us, even if we had the capability to leave Earth now at light speed. The amount of resources accessible to future humanity is shrinking every day as more and more galaxies move outside of our future light cone.
the non-zero probability that the universe can support an infinite number of computations means that the expected number of computations we expect to be performed in our universe is infinite.
Where do you get the non-zero probability from? If it’s from the general idea that nothing has zero probability, this proves too much. On the same principle, every action has non-zero probability of infinite positive utility and of infinite negative utility. This makes expected utility calculations impossible, because Inf—Inf = NaN.
I consider this a strong argument against the principle, often cited on LW, that “0 and 1 are not probabilities”. It makes sense as a slogan for a certain idea, but not as mathematics.
I’m not aware of any problems for “0 and 1 are not probabilities” if
Only applied to statements with short descriptions (otherwise they can’t be “picked out in advance” and it seems fine for e.g. an ML random infinite sequence of coin flips to have 0 probability), and of course the empty set gets 0 probability etc.
Your decision theory requires bounded utility functions, which tends to be an analytically useful assumption.
Here are examples of short statements with probabilities respectively 0 and 1: “0 = 1” and “1 = 1″. Or a little less trivially, the uniform distribution on the unit interval assigns 0 probability to each value in that interval and 1 to the whole interval. There is a class of theorems that the answers to certain probability questions about stochastic processes can only be 0 or 1.
I don’t think there is a good justification for bounded utility functions, any more than for supposing the real numbers to be bounded. It’s as artificial as supposing 0 and 1 to actually be not probabilities, and it does not solve any problems anyway.
yes, I know that the uniform distribution assigns probability zero to every point, that doesn’t mean I ever have to, which is why I specified ML random reals in the unit interval (Equivalently binary sequences). im well aware that many events have probability zero under some particular stochastic process (including the 0-1 laws), but in order to do that calculation you have to arrive at that stochastic process as a model and how will you assign it probability 1 in the first place?
The statement “0=1” is equivalent to the empty set, so also already covered in what I said.
In fact, assuming that the utility function is bounded solves many problems, such as pascal’s muggings and the saint Petersburg paradox. For history based RL, assuming continuity can also be useful, implying both boundedness and the existence of optimal policies.
Although this is fighting the hypothetical, I think that the universe is almost certainly infinite because observers such as myself will be much more common in infinite than finite universes. Plus, as I’m sure you realize, the non-zero probability that the universe can support an infinite number of computations means that the expected number of computations we expect to be performed in our universe is infinite.
As Bostrom has written, if the universe is infinite then it might be that nothing we do matters so perhaps your argument is correct but with the wrong sign.
Forget the erroneous probabalistic argument: it doesn’t matter if the universe is infinite. What we see of it will always be finite, due to inflation.
I think you mean lightspeed travel ?
That doesn’t rule out infinite computation, though, since in an infinite universe we have a perpetually increasing amount of resources (as we explore further and further at lightspeed).
No, I was referring to inflationary space-time. The fact that the universe is still expanding (and accelerating in its expansion) means that 92% of the observable universe can never be reachable by us, even if we had the capability to leave Earth now at light speed. The amount of resources accessible to future humanity is shrinking every day as more and more galaxies move outside of our future light cone.
Where do you get the non-zero probability from? If it’s from the general idea that nothing has zero probability, this proves too much. On the same principle, every action has non-zero probability of infinite positive utility and of infinite negative utility. This makes expected utility calculations impossible, because Inf—Inf = NaN.
I consider this a strong argument against the principle, often cited on LW, that “0 and 1 are not probabilities”. It makes sense as a slogan for a certain idea, but not as mathematics.
I’m not certain of this, but my guess is that most physicists would assign much great than, say, .0001 probability to the universe being infinite.
I’m not aware of any problems for “0 and 1 are not probabilities” if
Only applied to statements with short descriptions (otherwise they can’t be “picked out in advance” and it seems fine for e.g. an ML random infinite sequence of coin flips to have 0 probability), and of course the empty set gets 0 probability etc.
Your decision theory requires bounded utility functions, which tends to be an analytically useful assumption.
Here are examples of short statements with probabilities respectively 0 and 1: “0 = 1” and “1 = 1″. Or a little less trivially, the uniform distribution on the unit interval assigns 0 probability to each value in that interval and 1 to the whole interval. There is a class of theorems that the answers to certain probability questions about stochastic processes can only be 0 or 1.
I don’t think there is a good justification for bounded utility functions, any more than for supposing the real numbers to be bounded. It’s as artificial as supposing 0 and 1 to actually be not probabilities, and it does not solve any problems anyway.
yes, I know that the uniform distribution assigns probability zero to every point, that doesn’t mean I ever have to, which is why I specified ML random reals in the unit interval (Equivalently binary sequences). im well aware that many events have probability zero under some particular stochastic process (including the 0-1 laws), but in order to do that calculation you have to arrive at that stochastic process as a model and how will you assign it probability 1 in the first place?
The statement “0=1” is equivalent to the empty set, so also already covered in what I said.
In fact, assuming that the utility function is bounded solves many problems, such as pascal’s muggings and the saint Petersburg paradox. For history based RL, assuming continuity can also be useful, implying both boundedness and the existence of optimal policies.