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

## Paradox Tuesday – Tennis Ball Paradox

Posted by Maiu
on April 2, 2013

Posted by Maiu
on April 2, 2013

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

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.

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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.

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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.

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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.

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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.

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