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).
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.