Infinity Paradox or just confusion

  1. Given infinite time, can one do infinite tasks an infinite amount of times?

    How many times can one do infinite tasks in infinite time? One would think the ansewr to that question could not be >1, for that would mean that the tasks must have an end and all of them can eventually be completed (implying that they are finite in number, which we said they are not).

    Now you must consider this: if you were to pick any member of the set of infinity^infinity, I could always pick a member of the set of infinity with a higher numerical value. Doesn't this mean that infinity always equals infinity multiplied by any number? (right now I'm in pre-calc, and the concept of infinity hasn't been explored much yet, so forgive me if there is something I don't understand)

    Now with that information, we go back to the original problem, that if infinite tasks can be completed more than once in infinite time, then our assumption that the tasks are infinite in number is contradicted. We said that we should be able to complete infinite tasks in infinite time only once, and no more. But according to the 2nd paragraph, wouldn't completing infinite tasks once be the same as completing infinity^2
    tasks? And then wouldn't that be the same as completing infinite tasks infinite times? If so, then there is a contradiction.

    or is there something I am missing...?
  2. jcsd
  3. DaveC426913

    DaveC426913 16,439
    Gold Member

    This is why you can't do math with infinity. Infinity is not to be treated as if it is simply a regular number, even if one of very, very large magnitude.
  4. ahh I see. So that would just be another paradox showing WHY you can't treat infinity as a number then. right?
  5. DaveC426913

    DaveC426913 16,439
    Gold Member

    Yyyyeahh. .. but it's not like mathematicians said "if we do this with infinity we'll get weird results, so let's not."

    Infinity is defined, and does not mix with the real numbers.
  6. Sure you can do an infinite number of tasks twice in an infinite ammount of time.

    You can do this:

    Task 1
    Task 1 again
    Task 2
    Task 2 again
    Task 3
    Task 3 again
    ... again
  7. CRGreathouse

    CRGreathouse 3,682
    Science Advisor
    Homework Helper

    You can do a countably infintie number of tasks a countably infinite number of times. Just use a diagonal argument:

    1. Task 1
    2. Task 1
    3. Task 2
    4. Task 1
    5. Task 2
    6. Task 3
    7. Task 1
    8. Task 2
    9. Task 3
    10. Task 4

    You can see that every task appears an infinite number of times.

    It's also not hard to show that an uncountable number of tasks can be done a countably infinite number of times in uncountable time using digit mangling on the unit interval.
  8. CRGreathouse

    CRGreathouse 3,682
    Science Advisor
    Homework Helper

    You aren't talking about infinity ^ infinity, you're talking about infinity ^ 2. A (cardinal) infinite number squared is itself, and a (cardinal) infinite number times any positive finite number is itself.
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