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I have read many defferent versions of the Ross-Littlewood Paradox.

This post: Fun quiz: where did the infinitely many candies come from?

This post: Paradox: increasing sequence that goes to $0$?

This post: A strange puzzle having two possible solutions

And many others.

In all of them, a great effort is made to note that actions are performed in decreasing time intervals (1/2 second, 1/4 second, 1/8 second...). I am wondering why this specification is so important to the paradox? I understand that it stops the infinite steps from taking infinite time. The thing is the steps must then be performed infinitely fast.

Why is it that performing actions infinitely fast is so much more believable than performing actions for an infinite amount of time? In my opinion the latter is more plausible. Also why is believability so important for a paradox which is clearly impossible to execute?

Edit: I also wanted to note that there are similar things where we don't seem to need this kind of action. For example the Infinite Monkey Theorem. Why is it important in one and not the other?

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  • $\begingroup$ It's not important. It's part of the story. $\endgroup$
    – Asaf Karagila
    Commented Jul 29, 2014 at 14:43
  • $\begingroup$ Yet I have heard so many different versions. None of which make any mention of the original, and all of which take a lot of effort to point it out. $\endgroup$ Commented Jul 29, 2014 at 15:06
  • $\begingroup$ Also the fact that supertasks is entire tag on its own kind of shows its importance. $\endgroup$ Commented Jul 29, 2014 at 16:14
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    $\begingroup$ I can start tagging things with asaf-likes-this, will it make it important? $\endgroup$
    – Asaf Karagila
    Commented Jul 29, 2014 at 16:39
  • $\begingroup$ @AsafKaragila: I am intrigued about learning about foundations, so it would make them important to me if not important period. $\endgroup$ Commented Nov 28, 2015 at 8:05

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The time aspect is significant here because it introduces the concept of the "omega point" under the radar. That is, the questions are about the state of a system after a supertask has been completed. If each step takes the same amount of time, it is not clear that there is such a thing as "after the task is finished". To state the problem, you then need to introduce some idea of "time after the end of infinite time", which opens up a whole keg of worms that distract from the puzzle itself.

By stating the problem as one of exponentially increasing speed leading to finite time, then even though one is still expected to accept the completion of infinite tasks, the concept of state after the tasks are completed becomes intuitively understandable. One does not have to get into discussions of infinite numbers. This allows the person questioned to concentrate on the central issue.

Of course, you really have introduced numbers higher than infinity in the 2nd approach, which is completely equivalent to the first. But you've done it in such a way that often people don't even realize that it has occurred.

As for the Infinite Monkey Theorem, it does not require any concept of supertasks or omega points to understand. It states that almost surely if a random stream of symbols is produced long enough, a given specific string will occur in it. This is something that occurs in finite time, not infinite.

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