This doesn't really affect anything here at all. It may be neat that they're predicting time to end over 3 billion years from now, but I can guarantee we'll all be long dead by then.
Time Likely To End Within Earth's Lifespan, Say Physicists
31–40 of 64 posts
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#32Earlier quoted context omitted.
No, that "integer" would need to have an infinite number of digits, and no integer has an infinite number of digits.
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. It is an irrational number. We can never "see" all the digits of pi, because no matter how far out we go there are always more digits.
With an integer, this isn't the case. There are infinite integers, but this property is defined on the set, not on any single element in the set of integers. Every integer (a single element in the set of integers) is finite. All of them, no matter what. They all have an end.
That said, what "arbitrarily long" means is that for any length you choose, there are integers that long (or longer). Say we choose a length 5, then 10000, 10001, 10002, ... all have length 5 or greater, but every one of them has an exact length. It just happens to be greater than 5. You can do this for any length: 1000, 10^1000, whatever. Anything. Choose any number as big as you can even imagine. Try to imagine numbers bigger than anything you can even imagine. There are integers that long, and infinitely more integers longer than that. But every single one of them has an exact, finite length.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#33"If the universe lasts forever, then any event that can happen, will happen, no matter how unlikely." That's just plain mathematically wrong. There are many kinds of infinity ( http://en.wikipedia.org/wiki/Aleph_number ), and the set of all possible events has a much, much higher cardinality than an infinite axis of time. This is similar to saying that, since the set of integers is infinite, it must contain Pi.
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.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#34Earlier quoted context omitted.
Less of the whole arXiv, or just the subset of the arXiv which is crazy?
The problem is that you need to be a domain expert to tell which parts are crazy. It's not a general consumption resource.
True. Though the best heuristic that a non-physicist can apply is that if something on the arXiv sounds interesting, it's probably crazy.
Non-crazy papers have titles like Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization and Towards finite density QCD with Taylor expansions and tend to be of interest to around four people in the world, of whom three may or may not be the authors.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#35"If the universe lasts forever, then any event that can happen, will happen, no matter how unlikely." That's just plain mathematically wrong. There are many kinds of infinity ( http://en.wikipedia.org/wiki/Aleph_number ), and the set of all possible events has a much, much higher cardinality than an infinite axis of time. This is similar to saying that, since the set of integers is infinite, it must contain Pi.
the set of all possible events has a much, much higher cardinality than an infinite axis of time. Are you sure of that? How would you prove it? It seems intuitively right, but intuition has a habit of being wrong, when it comes to infinities.
I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an infinite amount of time has passed (aleph-null, by definition). Have I seen every possible sequence?
Well, take the set of aleph-null length sequences (all aleph-null of them [1]) and put them in an order. Now construct the following sequence: invert bit 1 from sequence 1, invert bit 2 from sequence 2, ... etc.
Note that this sequence was never produced by the machine, since we enumerated all the sequences it produced, but they are all different to this one. So, by a standard diagonalisation, the set of events is higher cardinality.
[1] How many aleph-null length sequences come out of the machine? Well, they have to be continuous, so the only obvious aleph-null length sequence is the total sequence the machine produces. However, you can drop an entry from the beginning of the sequence to get another aleph-null length sequence. And so on, making a total of aleph-null.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#36Earlier 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?
Integers can have an arbitrarily large but finite number of digits. They cannot have an infinite number of digits. There is a big difference.
I guess we are in the realm of cardinality of infinity which I don't quite get. I would think an infinitely big integer would have an infinite number of digits. So you are saying:
log(infinity) < infinity
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#37Earlier quoted context omitted.
The problem is that you need to be a domain expert to tell which parts are crazy. It's not a general consumption resource.
True. Though the best heuristic that a non-physicist can apply is that if something on the arXiv sounds interesting, it's probably crazy. Non-crazy papers have titles like Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization and Towards finite density QCD with Taylor expansions and tend to be of interest to around four people in the world, of whom three may or may not be the…
Right, and "Eternal Inflation Predicts That Time Will End" totally fails that test. (I don't mean to sound like I'm attacking you -- I agree with you. This is re: the OP)
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#38Earlier quoted context omitted.
Integers can have an arbitrarily large but finite number of digits. They cannot have an infinite number of digits. There is a big difference.
I guess we are in the realm of cardinality of infinity which I don't quite get. I would think an infinitely big integer would have an infinite number of digits. So you are saying: log(infinity) < infinity
Once again, there is no such thing as an infinitely big integer. You seem to be having difficulty with the fact that:
1. There are an infinite number of integers, but
2. Every integer is finite
Start counting 1,2,3,4... Do you ever get to the end? No, this proves that there are infinitely many integers. But do you ever get to an infinite number? No, you'll stay in the set of finite integers forever.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#39Earlier quoted context omitted.
the set of all possible events has a much, much higher cardinality than an infinite axis of time. Are you sure of that? How would you prove it? It seems intuitively right, but intuition has a habit of being wrong, when it comes to infinities.
I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an infinite amount of time has passed (aleph-null, by definition). Have I seen every possible sequence? Well, take the set of aleph-null length sequences (all aleph-null of them [1]) and put them in an order. Now construct the following sequence: invert bit 1 from…
Interesting attempt, but I'm not convinced by the part of the argument where you divide the complete output of the flashy light box into aleph-null sequences, each of length aleph-null.
Mostly, I'm unconvinced that an "event" which takes an infinitely long time to complete actually counts as an event. Usually when we talk about events they're localised in space and time.
Re: Time Likely To End Within Earth's Lifespan, Say Physicists
#40Earlier quoted context omitted.
> This is similar to saying that, since the set of integers is infinite, it must contain Pi. Not exactly. Pi has no chance of occurring in an infinite set of integers.
Okay, so how about "Given a number generator that produces numbers for an infinite amount of time, it will produce Pi" This, of course, is impossible.
I think you're missing the point. While it's _extremely_ unlikely that the next random number produced would be Pi, in an infinite scale it can (and theoretically would) happen because it is, in fact, possible to occur.