Subtleties of infinity

[stevo247]

Recently, I had taken a look at some of the works of the Presocratic Philosophers, particularly Parmenides and Heraclitus. Basically, Parmenides thinks that the "One" is immobile and unchanging and that motion is an illusion. His student Zeno gives the following as a proof: Can Achilles with his winged feet reach the boundary stone of the stadium? He needs a lapse of time to travel half the distance, and still another lapse of time to travel half the remaining distance, and so on and so forth. Hence he needs infinitely many time intervals to reach the stone; but that is an infinite amount of time. Hence, Zeno stopped Achilles in his tracks.

 

In a book called Quantum Philosophy, I read "we are no longer troubled by this paradox, because we know that the sum of an infinite number of (unequal) time intervals may be finite. This example is nevertheless interesting, because it reminds us of the extent to which the the logical treatment of infinity is subtle."

 

"The sum of an infinite number of (unequal) time intervals may be finite". Can anyone help me get a handle on that?

[Amr Morsi]

If the Time Intervals are tending to zero, as time increases, then the summation MAY be finite.

You have for example the geometric sequence; the sum of infinite number of terms is equal to a/(1-r), where a is the first term and r=Term(n)/Term(n-1), but r must be smaller than 1 for the summation to converge.

[vordhosbn]

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter of a beer. Before the next one can order, the bartender says, “You’re all assholes,” and pours two beers.

 

[the tree]

vordhosbn's joke gives a pretty good example.

[math]1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots=2[/math]

As for why, try sketching it.