41–50 of 64 posts
Earlier 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.
Earlier quoted context omitted.
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.
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 prove that the set of real numbers is uncountable.
Earlier quoted context omitted.
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.
Proof: let X[n] be the set of numbers with non-zero probability of being produced at the n'th trial. X[n] must be countable, since sum(X[n]) = 1 and the sum of any uncountably infinite set of non-zero numbers must be infinite.
Let X = union(X[n], n=0...infinity). X is countable, being the countable union of countable sets. The reals are uncountable. Thus, most real numbers will NOT eventually be produced.
(It's true, pi in particular could be in X, but the vast majority of numbers could not be in X.)
Earlier quoted context omitted.
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.
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…
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 eventually happen (and indeed, happen an infinite number of times as the paper claims).
"Earth's lifespan" is slightly misleading if we're talking about inevitable future catastrophes of physics. What's more important is the lifespan of earth as a hospitable planet and that's significantly less time (since the sun is heating up). From http://en.wikipedia.org/wiki/Sun#Life_cycle "...the Sun is gradually becoming more luminous (about 10% every 1 billion years), and its surface temperature is slowly rising…
Or we could just move this rock.
"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.
> 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.
"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.
(In fact I suspect he can never visit more than a vanishing subset of them, but I could be wrong, and I'm sure somebody has figured this out already...)
Earlier quoted context omitted.
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.
No, it theoretically would not happen. Proof: let X[n] be the set of numbers with non-zero probability of being produced at the n'th trial. X[n] must be countable, since sum(X[n]) = 1 and the sum of any uncountably infinite set of non-zero numbers must be infinite. Let X = union(X[n], n=0...infinity). X is countable, being the countable union of countable sets. The reals are uncountable. Thus, most real numbers will…