Eh, I don’t think it’s irrelevant, I think it’s interesting! I mean, consider a new infinite monkey experiment. Take the usual setup - infinite monkeys, infinite time. Now once you have your output… Do it again, an infinite number of times. Now suddenly those near impossibilities (the almost surely Impossibles) become more probable.
I also think it’s interesting to consider how many infinite sequences there are which do/do not contain hamlet. This one I’m still mulling over… Are there more which do, or more which don’t? That is a bit beyond my theoretical understanding of infinity to answer, I think. But it might be an interesting topic to read about.
In terms of the question, “Are there more infinite sequences that contain Hamlet or more that don’t?”- in the context of true randomness and truly infinite sequence, this feels like almost a trick question. Almost every truly random infinite sequence will contain Hamlet an infinite number of times, along with every other possible finite sequence (e.g., Moby Dick, War and Peace, you name it). In fact, the probability of a random infinite sequence not containing Hamlet is effectively zero.
Where it becomes truly interesting is if you have an infinite number of infinite sequences. Now you’d certainly get instances of those “effectively zero” cases, but only in ratios within infinity itself, haha. I guess that’s probably what you were getting at?
I thought that at first… But then for every infinite series with exactly one hamlet in it, there’s an infinite series where one character is wrong. And there’s another one where a different character is wrong… And so on and so on. Even if the series contains an infinite number of hamlets, you can replace one character in each in a huge number of ways! It starts to seem like there are more options with almost Hamlet than there are specifically with Hamlet.
In fact, I begin to wonder if almost any constraint reducing the search space in the infinite set of such infinite sequences, you will inevitably have fewer items within the search space than without… Since you can usually construct multiple non-matching candidates from any matching one.
But… Honestly I’m not sure how much any of that matters in infinite contexts. Since they are impossible it boggles my mind trying to imagine it.
Eh, I don’t think it’s irrelevant, I think it’s interesting! I mean, consider a new infinite monkey experiment. Take the usual setup - infinite monkeys, infinite time. Now once you have your output… Do it again, an infinite number of times. Now suddenly those near impossibilities (the almost surely Impossibles) become more probable.
I also think it’s interesting to consider how many infinite sequences there are which do/do not contain hamlet. This one I’m still mulling over… Are there more which do, or more which don’t? That is a bit beyond my theoretical understanding of infinity to answer, I think. But it might be an interesting topic to read about.
Fair enough, I suppose it is interesting!
In terms of the question, “Are there more infinite sequences that contain Hamlet or more that don’t?”- in the context of true randomness and truly infinite sequence, this feels like almost a trick question. Almost every truly random infinite sequence will contain Hamlet an infinite number of times, along with every other possible finite sequence (e.g., Moby Dick, War and Peace, you name it). In fact, the probability of a random infinite sequence not containing Hamlet is effectively zero.
Where it becomes truly interesting is if you have an infinite number of infinite sequences. Now you’d certainly get instances of those “effectively zero” cases, but only in ratios within infinity itself, haha. I guess that’s probably what you were getting at?
I thought that at first… But then for every infinite series with exactly one hamlet in it, there’s an infinite series where one character is wrong. And there’s another one where a different character is wrong… And so on and so on. Even if the series contains an infinite number of hamlets, you can replace one character in each in a huge number of ways! It starts to seem like there are more options with almost Hamlet than there are specifically with Hamlet.
In fact, I begin to wonder if almost any constraint reducing the search space in the infinite set of such infinite sequences, you will inevitably have fewer items within the search space than without… Since you can usually construct multiple non-matching candidates from any matching one.
But… Honestly I’m not sure how much any of that matters in infinite contexts. Since they are impossible it boggles my mind trying to imagine it.