R. Srinivasan wrote: > 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. > > Regards, RS
Well, they may be counter-intuitive for some people. I agree the Banach-Tarski paradox is a rather surprising result. But the fact that the sum of an infinite series of positive numbers can be finite I don't find counterintuitive at all, myself. Do you really claim to have a formulation of analysis where this result is avoided? Can you tell me what it is?