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.
> The NAFL resolution of these paradoxes is given in Sec. 4 of > <http://arxiv.org/abs/math.LO/0506475> (see Remarks 14-16). Basically > open/semi-open intervals of reals do not exist in the NAFL version of > real analysis -- so the proposition that Achilles is confined to the > interval [0,1) fails and cannot even be stated. Secondly it is not > legal in NAFL to ask *how many* intervals (or reals) are present in the > super-class of intervals ([0,1/2], [1/2,1/4] ....[1,1]), because direct > quantifiication over reals (or intervals of reals), which are infinite > classes/super-classes, is banned. But there is a way to quantify > indirectly, as explained in my paper -- and represent intervals, etc. > as "super-classes" mentioned above. > > Regards, RS