Mike Kelly wrote: > R. Srinivasan wrote: > > Mike Kelly wrote: >>> [...] > > > > > > But the sequence of nested intervals does get "as close as you like" to > > > zero, if you wait long enough. Any point other than zero will be > > > "passed" eventually. > You are considering a process in time, where you are at a particular > > interval, and then pass on to the next smaller nested interval, and so > > on. If you asked the Greeks, they would have probably told you that > > this infinite process of sub-diviision can never be "completed" and > > you will "always" be an interval's length away from zero. Whereas you > > have assumed that you can somehow complete this process, and after > > completion, only the point zero is never passed. That precisely is the > > issue in Zeno's paradox. > > > > > Note carefully the difference between your approach and mine. I am > > considering the *totality* of nested intervals (which classical > > standard real analysis permits) and saying that not one interval in > > this totality has end-points "infinitely close" to zero. > > It is not necessary for any interval to have end-points "infinite > close" to zero (whatever that means) to decide what is in the > intersection. You're considering which real numbers are in every one of > the nested intervals. If you pick any real number other than zero, then > some intervals don't contain it, so they aren't in the intersection. So > zero is the only real number in the intersection. > > "intersection" has nothing to do with "end-point infinitely close" to > something.
I don't deny that there is a proof that zero is the only point in the intersection. And there is also a simple proof-by-contradiction that would rule out the possibilty of an interval being in the intersection. All I am saying is that there is another paradoxical way of arguing that an interval has to be in the intersection. And as Herb has pointed out in another post, there may well be another way of arguing that there should be nothing in the intersection -- I am personally not able to get this intuition yet (will have to study the theorems he has mentioned). You can deny these paradoxical intuitions and see no paradox -- then there would be no way of contradicting you within classical real analysis. Or you can look for an alternative logic where there may be a different way of eliminating these paradoxes at the cost of some restrictions, and yet do much of practically useful real analysis. I am pursuing this latter route.