On Sunday, July 7, 2013 11:26:34 AM UTC-7, Julio Di Egidio wrote: > "Zeit Geist" <email@example.com> wrote in message > > news:firstname.lastname@example.org... > > > On Sunday, July 7, 2013 2:59:23 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > > > > > >> The old story of the marbles in the urn. In the limit the urn is empty > > >> but the limit of number of marbles is infinite. If limits have to be > > >> calculated by the finite terms, like in mathematics,then this is a > > >> contradiction. > > > > > > You mean the Urn where every marble is eventually removed, > > > and at Noon the Urn is empty? > > > > That's indeed a patent logical invalidity, already enough to warrant a > > revision of standard set theory. >
Start with an Empty Urn. At t = 0, add marbles labeled 1 and 2. At t = 1/2, remove 1, and add 3 and 4. At t = 3/4 = 1 - 1/4, remove 2 and add 5 and 6
In general; At t = ( 1 - 1/( 2 ^ n ) )remove marble n, and add marbles 2n + 1 and 2n + 2.
You can, easily, use those equations to find times, t_a and t_r, when any marble is added and removed.
Now, this entire things seems logically invalid, And IS paradoxical. However, the logic is fine.
The "invalidity" comes into play in the Physical World. First, I can't, really, have an infinite number of marbles. And, even granted that wish, I don't have an Urn large enough to hold them all. Second, nothing, person nor machine, can double the rate of a task infinitely many times. Also, we have that whole quantification of Space/Time.
Now, ask yourself the follow, in a purely Logical sense, NOT Physically.
Can we fill an Urn with all the marbles, using a Supertask in which we add the marbles two at a time, and cut the time in half between each step?
Can we empty an Urn containing all the marbles, using a Supertask in which we remove the marbles one at a time, and cut the time in half between e.s.?
In the above Supertask, has each marble been added to the Urn before it needs to be removed?
If you answer "Yes" to those three questions, then "At t = 1, the Urn is Empty." Is Logically Valid.
If you don't believe that we can Mathematically divide an interval in half infinitely many time, or have All Natural Numbers, then you had better re-read all of WM's post; because everything he says follows from that.