Paradox Tuesday – Tennis Ball Paradox

Are you up for a real challenge? Find the flaw in this argument and let us know in the comments!

** Note: Some posts on Math-Fail are user-submitted and NOT verified by the admin of the site before publication. If you find this post to be distasteful, non-math related, ?or something worse?, then definitely leave a comment letting me know. Thanks very much! Mike **

1 Star2 Stars3 Stars4 Stars5 Stars (5.00 from 1 votes)


  1. Youtube doesn’t want me to comment today.

    Anyway, at 1:00 the narrator says “Thus, in the limit there will be infinitely many balls in the room”, without having established that there is a limit. And in fact, the subsequent argument proves that no such limit set exists.

    Thumb up 0 Thumb down 0

  2. It’s a meaningless question. One cannot talk about “stage infininty”. One can say that at stage N, balls N+1 through 2N (or N balls) will be in the room. As N approaches infinity, the number of balls in the room will also approach infinity.

    Thumb up 0 Thumb down 0

  3. It makes no sense to ask what happens after a process with an infinite number of discrete steps has been completed. It is not possible to complete a process with an infinite number of discrete steps.

    Thumb up 0 Thumb down 0

  4. Otto Bob Otto

    There seems to be some major misunderstandings about infinite limits in general, and many of the videos/proofs play fast and loose with that.

    The real idea behind a limit going to infinity is that, in this case, the number of tennis balls in the room can be made arbitrarily large by repeating this process of enough times. It’s a concept that says the number of tennis balls grows without bound and this process is repeated. There are never an “infinite” number of tennis balls in the room, since a physical process cannot be repeated forever. We can have an arbitrarily large number of tennis balls, and wherever we stop, we CAN name which balls are in the room.

    Thumb up 0 Thumb down 0

  5. the number of balls goes to infty. The set of balls goes to the empty set. Taking cardinality does not commute with taking limits, apparently.

    Thumb up 0 Thumb down 0