Infinity

[Sarkus]

Can't you use this same this same logic on anything? There's always a longer finite number of times you can type 1, therefore you can't type 1 an infinite amount of times over an infinite amount of time. Sorry if I missed something, I only skimmed through your responses, if you already addressed this then go ahead and ignore me (busy atm).

You stated that given an infinite amount of time you could write out (or the monkey could type out) every possible finite sequence of letters an infinite number of times.

I pointed out to yourself, and swivel, that this is incorrect - as the series of finite sequences is unbounded, and thus infinite in length (i.e. the series is infinite in length although the individual elements are finite).

So given that a single monkey is unable to type out even once what can not exist... how could he do so an infinite number of times?

 

[swivel]

You simply can not write out an unbounded infinite (so as to differentiate it from the very different matter of bounded infinitessimals) series, even if every element of that series is finite.
And yet you seem to think you can not only do it once, but do it an infinite number of times.

I disagree. I understand your complaint fully, and I reject it as wholly incorrect.

It is true that you cannot complete an infinite set, but you can write out every member of a set bounded on one side. That is, you can complete a ray, but not a line.

I am granting the writer infinite time for the task. They have an infinite number of finite sequences to write out. Excellent! They have just the right amount of time for the job.

Will they ever be done? No. Will they ever run out of time, so that you can say, "See, swivel, they didn't finish the job"? No. Every swquence will come and every one will be written down.

Hotel rooms are often used to compare the sizes of infinite sets in order to determine which infinite sets are smaller or larger than others. In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.

 

[swarm]

In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.

One minor problem. They aren't brute force doing every possible combination of characters. Each pass is randomly generated.

 

[swivel]

One minor problem. They aren't brute force doing every possible combination of characters. Each pass is randomly generated.

Which is where they get their extra-simian strength in these examples.

I mentioned earlier that the set of (things monkey types) and (all possible sequences) are the same size, but if you distinguish duplicates from one another, the typed set is a larger infinite set than the set of all possibilities. Because the monkey is going to type each possibility an infinite number of times.

Of course, you can just group each instance together and you still have a 1-1 ratio, but it is amazing to realize that their set is much larger, not much smaller as others have argued in this thread.

 

[swarm]

Its been way too long since I did serious math to know if things collapse tidely or not

 

[Sarkus]

I disagree. I understand your complaint fully, and I reject it as wholly incorrect.

It is true that you cannot complete an infinite set, but you can write out every member of a set bounded on one side. That is, you can complete a ray, but not a line.

I am granting the writer infinite time for the task. They have an infinite number of finite sequences to write out. Excellent! They have just the right amount of time for the job.

Will they ever be done? No. Will they ever run out of time, so that you can say, "See, swivel, they didn't finish the job"? No. Every swquence will come and every one will be written down.

Hotel rooms are often used to compare the sizes of infinite sets in order to determine which infinite sets are smaller or larger than others. In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.

Yes - it is a problem - as it was claimed that they could not only just do the infinite task once, but an infinite number of times.
Above you seem to agree that "Every swquence will come and every one will be written down." I agree that every sequence will be written down ONCE.
But an infinite number of times each? No. You would need an infinite number of hotels... or in the scenario originally posted - an infinite number of monkeys.

 

[Nin']

Yes - it is a problem - as it was claimed that they could not only just do the infinite task once, but an infinite number of times.
Above you seem to agree that "Every swquence will come and every one will be written down." I agree that every sequence will be written down ONCE.
But an infinite number of times each? No. You would need an infinite number of hotels... or in the scenario originally posted - an infinite number of monkeys.

If you agree that every sequence will be written down once then don't you concede to my statement? If it can be written even once that means that it can be written again, however improbable. Since we're talking about infinity here probability is irrelevant, we just need to discern whether or not there is a chance. Since there is a chance, it will be written infinite times, but the more improbable ones will simply just appear less.

 

[ancientregime]

I read an interesting question and i was wondering what you all thought about it.

Hypothetically, if a monkey were to type randomly on a typewriter for infinity, would at some point the complete, exact works of Shakespeare be typed?

Depends on his genetic code, which is determined, and the how the environment (determined) surrounding him interacts for the rest of infinity. If he doesn't get caught in a perpetual energy loop that was uninterrupted for the rest of eternity, yes. Otherwise, no.

I got one to add, considering no perpetual energy loops can exist (keeps things in a permanent stasis). What are the chances that this present world we know exists? Well, we know it at least happened once. What are the chances it happens again? Well, considering eternity and no perpetual energy loops, it will happen. In fact, with no perpetual energy loops, it will happen not only exactly the way it did, but it will happen every way infinitely possible. It will be like ground hogs day to an infinite power.

 

[Ladicius]

I have a question about the monkey/Shakespeare thing. (tired so forgive the halfassedness....). To my understanding, Infinity isn't a number, and it can't be measured; while time is a unit of measurement. Hmmm, wasn't much of a question after all. Gn. School in a few hours.

 

[Sarkus]

If you agree that every sequence will be written down once then don't you concede to my statement? If it can be written even once that means that it can be written again, however improbable. Since we're talking about infinity here probability is irrelevant, we just need to discern whether or not there is a chance. Since there is a chance, it will be written infinite times, but the more improbable ones will simply just appear less.

No - I don't concede .
You claimed an infinite number of infinite length sequences.
I claim that the single monkey can only ever write a finite number of infinite-length sequences, or an infinite number of finite-length sequences. But not both an infinite number of infinite sequences.

Your flaw is "If it can be written even once that means that it can be written again, however improbable." It will take an infinite time to write once. There is thus nothing else it can do.

 

[one_raven]

I think we are all aware that monkeys don't live forever, strawberryfyre, but thanks for the reminder.

One monkey or a hundred or a million makes no difference. More monkeys just means parallel processing. Stuff will be typed a bit faster, that's all. The same "final" document will still be produced, in the long (infinite) run.

I disagree.

Just because the patterns are "random" (which is impossible, really) does not mean that eventaually every possible pattern would be produced.
Why would it?
Doesn't that presume that the monkey can not repeat himself?

If you disallowed repeating patterns, it would definitely not happen, because I am fairly certain Shakepeare used the work "King" more than once.
The only possibly way to ensure that it "will" be produced is to set restraints that A.) a "pattern" is a specific number of characters and spaces which is the resultant text (let's say you were going for Moby Dick, and Moby Dick had 234,583 characters the pattern size would be 234,583) and B.) that pattern could not repeat itself.
Without those two constraints I don't see how anything MUST be produced.

Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.
Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.

 

[swivel]

Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.
Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.

Yes, we can prove this for an infinite length of time. It would be impossible for the monkey to NOT pound out Shakespeare's entire cannon, punctuation and all.

 

[one_raven]

Yes, we can prove this for an infinite length of time. It would be impossible for the monkey to NOT pound out Shakespeare's entire cannon, punctuation and all.

I have seen no proof.

 

[Spectrum]

"I say no. How can an infinite amount of randomness produce a pattern? There can be no pattern, or it's not random." Excellent work Alpha

 

[draqon]

Aren't strings energy densities random, yet they produce a pattern...of quantum particles...atoms...molecules...

 

[Sarkus]

I disagree.

Just because the patterns are "random" (which is impossible, really) does not mean that eventaually every possible pattern would be produced.
Why would it?

Because of the nature of randomness and infinite time.
Given an infinite amount of time, if something does not happen then it is not possible. If there is ANY chance of it occurring, however remote, then it WILL happen.

e.g. if you have a coin where you think that the result is random, if it never landed on TAILS in an infinite amount of time then the result is NOT random. Period.
If it is truly random then as the number of instances tends to infinity, the outcomes approach the probability function of that randomness.
E.g. if you toss a coin an infinite number of times, the results will be 50:50 Heads/Tails.
If it doesn't then the assumption of the nature of the randomness needs revising.

So if you accept the assumption of randomness - e.g. a flat probability distribution across the 50 or so characters (letters + punctuation etc) - and there is a probability of hitting the keys in the right order to produce Shakespeare's complete works... then it WILL happen.

Doesn't that presume that the monkey can not repeat himself?

No. Repetition is permitted and expected, but if he repeats himself so as to disclude the possibility of other combinations then his typing is NOT random.

Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.

Yes you can.
But you have to accept the assumption of randomness and understand what it means - i.e. that as the number of instances of an event tend to infinity (e.g. tapping of keyboard) then the outcome tends to the probability function of the randomness.

If we assume a flat probability function (all 100 characters of so, - letters, punctuation, including CAPS - having equal chance of being typed) then given an infinite amount of time each character would be hit 1% of the time. Not only that, but in every subdivision of the overall result you would expect the same distribution. The larger the sample, the closer to the probability function you get.
Also, if you take every-other-letter then this will have the same probability function... as will every-fifth-letter... every-100th-letter etc... because they are all infinite in length.

Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.

It is impossible NOT to type "blue" given an infinite time.
It is also impossible NOT to type the the complete Moby Dick, given randomness and given infinite time.

If you think it is possible that, in the entire infinite time, the monkey only ever types out the letter "A", for example, then there is no randomness.


If there is a chance of occurring, given an infinite time and the same conditions then it WILL occur.
The only things that won't are those that are impossible due to those conditions, or become impossible due to changing conditions.

Therefore if a certain combination in an infinite random sequence is possible, then it will occur.
And if a certain combination in an infinite random sequence is impossible then the sequence is not as random as you thought - and the assumptions need revising.

 

[cosmictraveler]

A picture made with randomness, fractals.

 

[swivel]

Thanks, Sarkus. I am out of town and hate that I can't find the time to respond to some of my favorite threads. I really enjoyed your post.

 

[swarm]

Given an infinite amount of time, if something does not happen then it is not possible. If there is ANY chance of it occurring, however remote, then it WILL happen.

Time is not known to actually be infinite. I'm also not sure that both your conclusions are supportable. They both make the assumption that all events which can happen have a discrete, independent chance of happening at every instant. That seems an unwarranted assumption.

 

[Sarkus]

The exercise assumes an infinite amount of time... it is thus irrelevant whether time actually is infinite or not.

And the conclusions are logically sound. You have merely taken the quote out of context (i.e. we are discussing the monkey tapping at random for an infinite time - with the assumptions of A: infinite time, and B: random tapping).
Further in my post I illucidated, stated:

If there is a chance of occurring, given an infinite time and the same conditions then it WILL occur.
The only things that won't are those that are impossible due to those conditions, or become impossible due to changing conditions.

Perhaps you missed this part?

Normally after an event has occurred, the conditions will have changed so as to make a repeat of the event impossible, which is why in reality not every outcome occurs, and normally why only one outcome from all the possibilities occurs.

E.g. If you allow only one toss of a coin, each outcome (Heads/Tails) has a 50:50 chance. It lands on HEADS, but after the initial toss the conditions have changed so as not to allow a further toss.
The outcome of TAILS thus becomes impossible.

But the essence of probability is that as the number of identical events approaches infinite, the outcomes of those events approach the probability function of the event. Do you agree with this?

Therefore, if you assume the entire work of Shakespeare is 100 million characters long, and you take each string of 100 million characters as an event... and there is ANY possibility that the monkey will randomly tap 100 million characters in the right order... then given an infinite number of attempts at this 100-million character string the monkey WILL type it out correctly.

In such a scenario - if the monkey does not type out something you think is possible (however remote that possibility) then it was not actually possible, and your assumption of possibility was wrong.

I hope this clarifies.