Of course not 'really' broken since there are no systems of infinitely many particles containing points of accumulation in reality. For physicists it is very important to realize that 'actual infinity' is not always a harmless device to avoid irrelevant complications, but that it allows to model things that don't work in reality (such as making two unit-spheres from a single one by dissection and reassembling). The cited article is fun to read. Thanks for the citation.

Of course not 'really' broken since there are no systems of infinitely many particles containing points of accumulation in reality. For physicists it is very important to realize that 'actual infinity' is not always a harmless device to avoid irrelevant complications, but that it allows to model things that don't work in reality (such as making two unit-spheres from a single one by dissection and reassembling). The cited article is fun to read. Thanks for the citation.

@Ulrich Mutze

Yes, I agree with you that this "effect" is completely owing to the cardinality properties of the infinity... But the Banach-Tarsky paradox, You mentioned above, lies in a completely different plane, as it relates to so called "unmeasurable" geometric objects, contrary to those from mechanics or from electrodynamics, where the main tools are based om strongly measurable components.

Yet my main point of interest lied in modern quantum electrodynamics where completely similar "calculations" with infinite series are performed for obtaining a miraclous and almost exact experimental values of thin structure alpha=1/137, gyro-magnetic factor g=2.0023318361(10) (theoretical prediction) and its experimental value g=2.0023318416(13)! It is looking strictly fantastic - I mean, the obtained accuarcy!

Regards!

I'm stuck.

When I was a high school student I explained to myself the Zeno paradox in the follwoing way:

Indeed, Achilles will cross infinite number of points, but the total time he will spend to catch up with tortoise is finite, because the sum of (1+1/2+1/4+...) is finite (convergent progression). Hence, the paradox is resolved.

As for infitite balls, I cannot figure out an approach.... But I feel that there has to be a solution. Something wrong is there in the following statement:

"after the finite time that it would have taken the first ball to reach the point 0, had the

other balls not been in its way, every ball will have moved briefly, but then have been brought to rest."

EVERY ball brought to rest?

The problem is that we cannot identify the ball that will not be brought to rest...