Rupert wrote: > R. Srinivasan wrote: > > Rupert wrote: > > > R. Srinivasan wrote: > > > > Barb Knox wrote: > > > > > > > > > > Consider that calculus too started out as a half-baked theory laden with > > > > > paradoxes, and that one of the great mathematical achievements was to > > > > > put it on a rigourous footing. And indeed, set theory was one of the > > > > > important tools in that enterprise. > > > > > > > > The presently accepted foundations of the calculus still does not > > > > resolve Zeno's paradoxes. > > > > > > Why not? What's the paradox? > > > > The paradox can be stated in two ways. Consider the example of Achilles > > chasing the tortoise along a straight line at a constant velocity > > higher than that of the tortoise's (constant velocity). The paradox is > > that Achilles has to "complete" infinitely many opertaions to catch up > > with the torotoise -- from the starting point he sees the tortoise at a > > particular location ahead, and he first has to reach that location. > > When he reaches, he sees the toroise ahead at another location, and > > Achilles has to reach there. And ad infinitum. The Greeks thought that > > this was a paradox presumably because they viewed infinity as > > "potential", i.e., the infinitely many operations required to reach the > > torotoise cannot never be completed in finite time. > > > > >From the modern point of view the paradox can be stated as follows. > > How can infinitely many finite, non-zero, non-inifnitesimal intervals > > of reals sum to a finite interval (why isn't the sum infinite)? I..e, > > suppose starting from location zero, Achilles first reaches 1/2, then > > 3/4, then 7/8,...... Then Achilles covers the distance > > 1/2+1/4+1/8.....=1, where he catches up with the tortoise. The paradox > > is -- why isn't the sum infinite, given that there are infinitely many > > finite, non-zero and non-infinitesimal intervals being summed (assume > > we are using some standard version of real analysis). > > > > Well, it just isn't. I don't see any reason why it should be. You > haven't shown a contradiction in standard real analysis.
That is true. The same can be said of the Banach-Tarski paradox or many of the other paradoxes of classical measure theory. But these are paradoxes nevertheless, and highly counter-intuitive.