Infinity

Discussion in 'General Philosophy' started by Bigtraine, Dec 17, 2001.

  1. swivel Sci-Fi Author Valued Senior Member

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    Agreed. We wouldn't get the Complete Works of Shakespeare in the proper order and with exact punctuation, we would get an infinite number of copies of them. And an infinite number of copies with every play spelled out in reverse. And an infinite number of copies with the plays arranged in every order possible. Even copies with scenes from one play inserted into another.

    Think about the nature of infinity. If the "editor" in our analogy has not yet gotten the document he is after, he has forever to wait for it. Once he gets what he needs, he has forever to wait on the next one. Is the single transcription of a lone letter at the very end of the Complete Works enough to dissuade them from their Sisyphean task? Of course not, they have an infinite number of stabs to go.
     
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  3. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    Alas, not true.
    As there are an infinite number of finite-length sequences, you can never type them all out even once, let alone an infinite number of times.

    You couldn't even finish typing out all the finite-length sequences of just the letter "a" (e.g. a, aa, aaa, aaaa, aaaaa,... ) due to there being an infinite number of them. Even if you used "aaaaaaaaaa" to be 1x 10"a" sequence and 10x1"a" sequence at the same time (and all permutations in between), you still wouldn't achieve it.
    You'd be left tapping away at "a" for eternity, never to succeed.

    I'd agree with this, because there aren't an infinite number of combinations for Shakespeare's complete works, but very much a finite number.
    So in an infinite time we would get an infinite number of this particular finite-length combination.

    But this is different to saying we would get every possible finite-length sequence, as there are an infinite number of those.
     
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  5. swivel Sci-Fi Author Valued Senior Member

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    I disagree. I would argue that we could get every finite-length sequence, as well as every sequence of any chosen length.

    This is the same distinction that I bring up above on the nature of the infinitesimal. As soon as you choose your finite length, we can state that every sequence will be arrived at an infinite number of times. If one tries to continue an eternal process of adding one more to that length, they will keep avoiding the inevitable, and thereby fool themselves into thinking they have escaped the simple fact:

    No matter which length you settle on, we can agree that all sequences are produced an infinite number of times. Every time you lengthen the sequence by one, this still holds. It will always hold. That is an eternity of correctness, not an eternity of defeat.

    Don't mistake the ETERNAL process of increasing the length with the INFINITE. You will never reach the infinite, even if you go on FOREVER in this manner. And at every step along the way the fact remains that you have a finite number in your sequence, and all possible outcomes will be produced.
     
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  7. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    But how do you reconcile that to there being an INFINITE number of finite-length sequences. And there are an infinite number of them.

    I agree that once you have chosen your finite length you will get an infinite number of them... but it is in choosing ALL finite lengths that is problematic when there are an INFINITE number of them.

    If you think there are a finite number of finite-length sequences, what is the largest finite-length sequence?

    Bear in mind that I agree with your comments regarding Shakespeare's work: it is specifically with Nin''s comment: "I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times. " that I disagree.


    To argue it another way:
    In an infinite pile of pebbles, what is the largest finite number of pebbles you can have?
    The answer is that there isn't one.
    Or do you disagree?

    If you agree, then how can you think you can reach the end of the infinite number of finite-length sequences to be able to repeat it, let alone an infinite number of times?



    It is a flawed comparison, though.
    The bounded range, no matter how infinitessimally small the interval, is very different when compared to an unbounded range.
    A bounded range can be leapt across with simplicity by something that is larger than the range.
    The unbounded can not - as there is nothing larger.
     
  8. swivel Sci-Fi Author Valued Senior Member

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    2,494
    The same way I imagine a stack of an infinite number of Collected Works of Shakespeare.

    The stack of infinite number of sequences of ever-growing size are a stack of FINITE length sequences, all of which would result in every combination possible.

    Are you ever going to add one more to the length of a sequence and confound the monkeys? Nope. So you have an infinite collection of an infinite collection. All of which obey a simple principle of statistics.
     
  9. John Connellan Valued Senior Member

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    3,636
    I had agreed with you that you will never reach zero or infinity.

    My other sentence was worded wrongly:

    I meant to say

    Funny thing is, the number of divisions you perform on this segment can be infinite!!!

    rather than

    Funny thing is, the number of divisions you perform on this segment can be infinity!!!

    and you had agreed with this by stating that
    which essentially means the same thing
     
  10. swivel Sci-Fi Author Valued Senior Member

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    2,494
    Gotcha. The power of a single letter, eh my fiend?

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  11. Nin' Registered Member

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    So which sequences won't be typed over an infinite amount of time and why? This implies that only a finite number of sequences will be written over an eternity which is false.
     
  12. swarm Registered Senior Member

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    The complete works of Shakespear are a finite sequence of characters.

    Starting from the big bang it took about 16 billion years to burn hydrogen down into heavier elements, form a planet, evolve monkeys, have them develop writing and completely at random write the complete works of Shakespear.

    Unfortunately they did it before developing the typewriter so as soon as the universe breaks back down into quarks we will have to start over and try again.

    Stupid monkies.
     
  13. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    There are an infinite number of finite sequences... so one monkey could never type them all out an infinite number of times.
    One monkey can only type out a finite number of things an infinite amount of times, or an infinite number of things a finite number of times.

    To write out an infinite number of things an infinite number of times would take an infinite number of such monkeys.


    How long is the longest finite sequence?
    Ironically it is infinitely long (i.e. there are an infinite number of finite sequences - with the longest being infinitely long).

    If one monkey starts by writing the longest finite sequence, it will take an infinite amount of time. He will be unable to write anything else.

    If he wants to write it an infinite amount of times, he could start, for example, writing the first letter of the sequence an infinite amount of times... but this lone monkey is then again unable to do anything else, such as move on to the secong letter in the sequence.


    A lone monkey, given an infinite amount of time, can either write a finite sequence an infinite amount of times, or an infinite sequence a finite amount of times. He can not write an infinite sequence an infinite amount of times.


    If you think differently I would say it is because you are assuming that because you have written "longest finite sequence" it must therefore be finite, and assume that it is thus not infinite in length.

    If it is not infinite in length, how long is it?
    Answer me that, and I might be able to see your / swivel's side of the argument.
     
  14. swivel Sci-Fi Author Valued Senior Member

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    2,494
    That can't be correct. The longest FINITE number can't be INFINITE. You are mistaking, once again, the eternal process of writing out finite numbers for the existence of an infinite number of them. True, you can continue making finite sequences as long as you like, but you never get to say that any of them go on forever.

    Since the monkeys type for an infinite length of time, they easily keep up with the process, typing out every possible sequence an infinite number of times.

    Going further, if you have an infinite number of monkeys that type randomly, but also for a random length of time, every sequence will be typed out perfectly, from beginning to end.

    And if you have an infinite number of monkeys with infinite time, each sequence of INFINITE length will also be typed out faithfully.

    There is nothing those smelly bastards can't do if you give them the power of infinity.
     
  15. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    10,353
    Then how long is the longest finite number?

    Let's just start with that question:
    How long is the longest finite number?

    Imagine it another way... imagine an infinitely large cake. What is the biggest chunk of this cake you can take and still leave some left.


    No, I'm not.
    Your confusion lies in you assuming that "the longest finite number" is finite in length - but it isn't. There is no "longest finite number":

    Proof:
    Imagine a finite number with X digits.
    There exists a longer number with X+1 digits, correct?

    Now imagine that the "longest finite number" has Y digits, and we know that Y > X.

    However, there exists a longer number than the one with Y digits, and it has Y+1 digits.

    There is therefore NO "longest finite number" - 'cos as soon as you state what it is, someone can add an additional digit - ad infinitum.

    Therefore there are an infinite number of finite numbers, and a single monkey can thus NOT type them all out an infinite number of times.


    Where does my logic break down?
    I have shown yours to break down in your assumption that the longest finite number is finite.



    As for the rest of your post, I have no issue with an infinite number of monkeys doing what you propose, but the OP is regarding a single monkey, or at most a finite number of monkeys.
    What you claim is logical for an infinite number of monkeys - just not a finite number.
     
  16. gluon Banned Banned

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    512

    Yes, because you would have an infinite amount of time to let the dirty little monkeys ''do their thing.''

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  17. John Connellan Valued Senior Member

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    Wow, this thread is wrecking my head!!!
     
  18. swivel Sci-Fi Author Valued Senior Member

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    2,494
    There is a problem with your proof. You already defined 'Y' as the "longest finite number". You can not then add '1' to this number two steps later.

    And I understand the difference between my scenarios and the OP's. I'm just having fun rolling the ideas around. I don't see this as a debate to get testy over. I'm certainly not worked up.

    This all boils down to your contention that there is a FINITE number with INFINITE size. I call bollocks. There is no END to the number of FINITE numbers, but each and every one is FINITE. The HYPOTHETICAL nature of their eternal mystery is what we call the infinite.

    Really, it is quite obvious that your reasoning fails since it is paradoxical. There is no finite number of infinite length. Please observe the lunacy of this statement.

    Again: There are an infinite number of finite-length numbers. There is no end to them. But each one has the length of x+1. None of them will ever NOT be calculable by that simple formula.
     
  19. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    Firstly if you re-read closely you'll note that I state "There is no 'longest finite number'". You apparently seem to concur. :shrug:
    All I have been doing, which you so aptly highlight as well, is raise the paradoxical nature of the assumption that there is a "largest finite number", and in doing so it makes the conclusions invalid.

    The very phrase "largest finite number" is meaningless.
    It has no place in logic, since logically it does not exist.
    Thus to use its existence as an assumption - or a claim - makes the conclusion / claim meaningless.

    You are trying to make a set of all finite numbers, when such a set can not exist precisely because of its paradoxical nature.
    To then say that someone can type out this set an infinite times is, thus, meaningless.

    Excellent - moving on then, you admit that there are an infinite number of finite-length numbers.

    Okay, next question... how long will it take one monkey to type out the infinite number of finite-length numbers just once, given that, as you put it "there is no end to them"?

    And next you'll be able to answer how one monkey, while spending an infinite amount of time trying to type out all the finite-length numbers, can then write them each an infinite amount of times?


    Glad to hear it.
    Just as long as you stick to the single monkey for the purposes of discussion with me - as I concur with your assessment of the type-writing ability of an infinite number of monkeys given an infinite amount of time.
    It is just this one solitary monkey that appears to be causing the issue, and his claim that, given enough time, he can do the same thing as the infinite number of his simian-colleagues can do. And, as you so succinctly put it earlier, I call bollocks on that.

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  20. swarm Registered Senior Member

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    There is no longest finite number. For any squence of numbers of n length there is a sequence n+1.

    Note n and n+1 are unbounded, but not inifinate. An infinate sequence can't be enumerated.

    If there is a finite probability of an occurance then it has that probability for each trail. I.E. if the odds are 1 in 6 then every throw they are 1 in 6 no matter what has happened in previous throws. So even though the odds are 1 in 6 and you make 6 throws there is no garranty you will get a hit. There is no garranty you will get a hit with a hundred throws either, but it is likely you will.

    With a finite probablility task, like the complete works of Shakespear, and an unbounded time frame, you are basicly saying the monkies will keep pecking away until they succeed. In this case it took about 16 billions years.
     
  21. Nin' Registered Member

    Messages:
    90
    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).
     
  22. swivel Sci-Fi Author Valued Senior Member

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    2,494
    Agreed. There is no longest finite number. But none of them are infinitely long. The process goes on forever (an eternity) but never gets where it is heading (infinity).

    That's why we can treat any "very large" finite number as the longest. Choose one. Treat it as if it is the "longest" and talk about it. Whatever you say about it will be true of the +1 number, and the one after that, and the one after that.

    Sarkus is making the mistake of treating this eternal process as if it is an infinite process. It is the same as with the infinitesimal. Pick a very big number or a very small number and call it the "last one". Settle on a metric and calculate away. If someone wants to move your metric, recalculate. Then analyze of the movement of your answers (or stasis in some cases).
     
  23. Sarkus Hippomonstrosesquippedalo phobe Valued Senior Member

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    It is you making the mistake of assuming that just because you can arbitrarily label a "very large" finite number as "the largest" and "finite" that this means you can write them all.
    If you admit that there is no largest finite number - how on earth do you propose to be able to write them all out when you know, as you have admited, that as soon as you think you have done so there is another, and another, and another that are longer.
    The numbers themselves might be finite - but because there are an infinite number of them you can not write them all out. It is not the numbers themselves, but the series as a whole you are seemingly ignoring.
    You seem to ignore this point.

    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.
     

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