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[SilentAvenger]

thebruce,
I will now quote the the zero-one law:
"a certain type of event, called a tail event, will either certainly happen or certainly not happen; that is, the probability of such an event occurring is zero or one."

Where the definition of tail event is "an infinite sequence of independent random variables (not necessarily identically distributed). Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subsequence of these random variables."

Finding a finite string inside an infinite one is a tail event, therefore the chance for each finite string to be found is either 0 or 1.

[Shish]

I like how everyone ignores me ._.

To rephrase my proofs as a challenge: if this discussion gets anywhere closer to a scientifically provable answer than the "infinite force vs infinite wall" one, I'll eat my hat.


A note on personality: although I'm normally of the "everyone should be allowed to do what they want, except interfere with other people" persuasion, I can't help but insist that people follow the laws of logic - Internet science discussion that I've seen has been particularly poor recently, only the other day someone was claiming that incomplete fossil records proved evolution false, and thus creationism must be correct; I'm still annoyed from that, so pardon my harshness when I point out that this whole debate is rather silly, and you're all wasting your time by partaking in it :)

ED> The specific debate I'm referring to is "can you prove that an infinitely long random number contains any finite sequence, without brute force looking for it"

[thebruce]

edit: sorry Shish, I just find the discussion intriguing... and it's something at least to do until the next big syzygy puzzle no ill-will intended at all in the chat...

[Jay]

I, too, am usually pretty open about the kind of discussions we have ehre, but I have to agree with Shish.

I don't want this to go on for much longer. I think a lot of us have been looking too hard at these numbers for too long. How about a little break, guys?

[Olorin]

[Jay]

[neon snake]

I have to say, I'm with 'thebruce'. There's certainly no ill will intended, and its fascinating - I havn't thought this much about maths for years.

As for wasting time, I'd prefer to spend time chatting about this than sitting on my hands watching the codebreakers go! Unfortunately, I don't have a lot to contribute to the crowd currently trying to break the number under the semacode!

Plus, the avenger has just fried my brain - I'd never heard the 'tail event' theory. Please expand - are we saying the chances of a possible event over an infinite timeline are both definite and impossible (1 and 0) at the same time? Or have missed something?

[neon snake]

'Kolmogorov's zero-one law'

I think I got it.

The zero-one law says that the possibility of something happening, if there are an infinite amount of trials (that's the key, that theoretically you can keep trying) is either 1 or 0.

Monkeys/Shapespeare -

Take one monkey, one typewriter, and remove all real world consideration such as the typewriter jamming, someone forgetting the feed the monkey and the fact that given an infinite amount of time the monkey will evolve into the leader of a large superpower. Assume he hits keys at random, and indeed make all other assumptions necessary to make it possible, for the sake of the experiment.

Popular theory says he will type the works of Shakespeare (and indeed, someone mentioned this earlier as an example).

There is a chance (although stupidly small) that he will get it right first time. Since that chance exists, he will get it right at some point in infinity. Therefore the probability is 1.

Now, remove the x key from his typewriter. There is now no chance that he will get it right first time. Therefore the chance of him ever getting it right is 0.


I think what silentavenger was saying, and it passed me by, I must admit, is that in an infinite string, the chances of something occuring are either 0 or 1. Theres no such thing as 'it has a 40% chance of occuring' - we can state with absolute certainty that it's either in there, or it's not.

So, with infinite numbers -

If there is the smallest possibility that a string can occur, then it does occur - probability 1. If the string is, for example, '1268759223841558', but Tanner has removed all the '3's from the infinite number, then the probability is 0.


Avenger? Is that right?

[tanner]

spot on -- if it can happen it will -- if it cant it wont

[Olorin]

[Shish]

> spot on -- if it can happen it will -- if it cant it wont

so says sense, but what if a random number generator purely by chance never stopped outputting 1s? It is possible for such a thing to happen (if you can say for certain that it'll stop outputting ones, then it isn't random), and given an infinite number of RNGs it thus *will* happen (see the tail thing).

And once again, we come up to the battle of the infinite randomness vs infinite chance of a RNG generating a non-random looking sequence

ED> As said in PMs, this whole thing has been solved, and the answer is thus: there is no solution.

To expand and clarify - If you could make guarantees about the contents of the number, then by defenition it isn't random.

[thebruce]

yay! I feel good... it seems we've come to a consensus...

basically, the way it started was the claim the any sequence of numbers can be assumed to be found in e at some point with absolute certainty.

As I thought, that's incorrect. There is no absolute certainty. Even if the chance approaches certain, it will never reach it, therefore given an infinite amount of time, calculating a chance is pointless, because in the end (which doesn't exist) it will always either be or not be... 1 or 0...

cool

[tanner]

[SilentAvenger]

Yes.

Now, to answer that question we need to know if e is truly a random string of digits.

Also, we need to ponder another thing:

Say S is a finite string, and is NOT in e, therefore, all strings containing S (the subset of all the strings, which consist of a string A of any length (0 to infinity) concatenated to the string S and concatenated to a string B, also of any length (even one different than A's)) and all strings contained within S (assuming S doesnt have repeating strings within it, ((1 + n)n)/2, this includes S itself, n is S's length.), arent in e either.

Now, if S isnt in e, there is a group of infinity degree 1 strings that arent in e. The question is, can this happen. I dont know enough math to tell.

(If you want the proof as to why its infinity degree 1, give a shout out)

[tanner]

all finite seqences will appear somewhere in e --- but infinite seqences are more problematic --- one could ask (a very interesting question), does e exist within e?

edit -- the answer is no ---

do not believe intuition