Time Likely To End Within Earth's Lifespan, Say Physicists • Hacker News Board

Re: Time Likely To End Within Earth's Lifespan, Say Physicists

#61

Post by alphaBetaGamma » Tue, Sep 28, 2010, 9:07 PM UTC

jules wrote: post #52

alphaBetaGamma wrote: post #33

Earlier quoted context omitted.

To give a specific example: In three and more dimensions, if you start a random walk outside some sphere there is a non zero probability that the event "the brownian motion enters this sphere" never happens. Funnily, in dimension one and two this is not true.

Interesting. Are you talking about a continuous walk, or a walk on the integers? Is there an intuitive explanation for this? I get that in 1 dimension you will visit all points if you wait long enough, but two? And why does it change at 3?

This is for continuous walk.

As you say, in one dimension you will visit all the points.

In two dimension the random walk is dense, so you pass arbitrarily close to every point in the plane.

As the dimension increases beyond two, it gets less and less likely that you hit the target sphere. In some way, space gets bigger and bigger as the dimension increases.

Suppose the d dimensional "target" sphere is very small and very close to the starting point of the path. Let some time pass, and suppose that the Brownian motion does not hit the sphere and has increased by 1 in every direction. Then the distance to the sphere is sqrt(d), which grows unbounded as d increases: if you don't hit the target immediately, on average you will be further away from it in higher dimension. So the probability of touching the target after that diminishes as the dimension increases.

I don't have an intuitive explanation of why 3 is the critical dimension though.

Re: Time Likely To End Within Earth's Lifespan, Say Physicists

#62

Post by alphaBetaGamma » Tue, Sep 28, 2010, 9:13 PM UTC

"Any type of event that has nonzero probability will happen infnitely many times(...). This undermines the basis for probabilistic predictions of local experiments. If infnitely many observers throughout the universe win the lottery, on what grounds can one still claim that winning the lottery is unlikely? To be sure, there are also infnitely many observers who do not win, but in what sense are there more of them?"

The first sentence is wrong, as pointed out by zeteo, but the rest does not make any sense either.

Take continuous random walk starting at zero at t=0. There are an infinite number of paths that reach each value at time t=1. Yet we can say the the probability that the random walk is positive at t=1 is one half. And by "say" I mean give a solid mathematical definition of this fact, not some hand waving argument.

Re: Time Likely To End Within Earth's Lifespan, Say Physicists

#63

Post by mzl » Wed, Sep 29, 2010, 9:53 AM UTC

jaf656s wrote: post #32

asmithmd1 wrote: post #23

Earlier quoted context omitted.

So you are saying that their is a largest integer with a finite number of digits? Lets call that integer N, wouldn't 10*N have one more digit than N?

You are confused: yes there are infinite integers, but every single integer has a finite number of digits. Edit: (Several people have said the same thing so maybe I can add something else) I think another point of confusion might be the difference between "arbitrarily long" and "infinitely long". The number pi is infinitely long, a single integer can be arbitrarily long. What's the difference? Pi has infinite digits.…

This is all true, assuming one uses a standard model for the integers. Non-standard models[1] are more interesting, but I'm not sure if one could say anything about the number of digits of a non-standard number.

[1] http://en.wikipedia.org/wiki/Non-standard_model_of_arithmeti...

Re: Time Likely To End Within Earth's Lifespan, Say Physicists

#64

Post by Robin_Message » Wed, Sep 29, 2010, 12:10 PM UTC

hugh3 wrote: post #46

theli0nheart wrote: post #42

Earlier quoted context omitted.

Well, it's kind of a Catch-22. This proof is dependent upon the lemma that time is infinite. On the other hand, the set of events we're looking for is the one where time ends. Therefore by assuming the lemma we have no need for the proof. OTOH, if we're looking at another event that's not so confusing (one that's not the end of time), the proof holds water. It's a pretty sweet proof that I think is similarly used to…

No, what I'm saying is that an event which takes an infinitely long time to happen (ie an infinite series of one-second flashes) isn't a proper "event", which as used at least in relativity, is something that happens in finite space and time. But there's no need to argue about the definition of event. I'll say instead that I'm interested in knowing whether infinite time implies that all finite-length events must even…

Yes, sorry, there is definitely a theorem that if you have an infinite sequence, uniformly distributed over an alphabet, then the probability of seeing any finite sequence is 1. I think that, given some assumptions, you can model the universe as an infinite numerical sequence, so you should see all finite events.

However, the assumptions here seem tricky. Uniform distribution of probability seems unlikely, and if the universe is anything like the game of life, there are garden of Eden states with no predecessor state, which an evolving system cannot reach, even in infinite time.

Cardinality is indeed tricky. I had thought that all finite sequences would be greater than aleph-null, but apparently it isn't.