Eternal Inflation "Time Will End"

Kevin_Axion

This paper came out last week and looks very interesting but of course relies on M-Theory: http://arxiv.org/abs/1009.4698

SW VandeCarr

The argument the authors use is that if time is infinite, all events that are not prohibited are possible. From this observation the authors seem to conclude that if all possible events must occur and must occur an infinite number of times, then the method of assigning probabilities is "undermined." If I understand this argument correctly, it is simply wrong.

The point is easily made with the natural numbers. They are an infinite set. The set of all even numbers is also infinite; so is the set of every 10th number, every 100th number and every [tex]100^{100}[/tex] th number. Moreover each of these sets has the same cardinality. Nevertheless it is quite possible to assign the probability of a random number. We can say, for example, that numbers divisible by 100 are only one tenth as likely as those divisible by 10 when sampling from an arbitrarily large or even "infinite" set.

Calimero

I took a glance at paper, and although it is interesting, that was my first doubt also. How come that probabillity sort of works in our world? Maybe I don't understand something.

SW VandeCarr

Did you follow my argument in the second paragraph of my post? If not, I'll try to explain further. These are generally accepted concepts regarding infinite sets, on which probability theory is based. Note important distributions like the Gaussian are infinite but the integral of the probability distribution function is unity.

EDIT: I edited the first paragraph of my original post since some of the things I said were not explicitly expressed in the paper.

chronon

This is wrong. It is not possible to have a uniform probability distribution over the natural numbers.

SW VandeCarr

If you read my post, I'm not talking about the probability of selecting a given number. I'm talking about the probabilities of selecting numbers wholly divisible by 10, 100, 1000, etc. as examples. If you take finite random samples of size N from the set of natural numbers, the relative proportions of such numbers in the sample will tend to match the frequency in the set as N grows large. In probability theory this holds even if the set being sampled is infinite provided you can define the set by induction. So the probability of selecting a number divisible by 10 is 0.1.