Rupert wrote: > 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?
Another way to state the paradox is as follows. Consider the infinite series of nested real intervals [-1,1], [-1/2, 1/2], [-1/4,1/4],... The intersection of these intervals contains the single point 0, but *each* of these infinitely many intervals is of non-zero and non-infinitesimal length. So why doesn't their intersection contain infinitely many points? Again you may not find this counter-intuitive, but many do.
The NAFL version of real analysis is outlined in the reference [arxiv: math.LO/0506475] given in my previous posts. The result is "avoided" in the following senses -- the above infinite series of intervals, when represented as a super-class by the method outlined in that ref., *must* also include [0,0], i.e. the interval of zero length. So it is not possible to talk of infinitely many non-zero, non-infinitesimal intervals of the above series in NAFL. Secondly it is *not* valid to ask the question "How many intervals are present in the super-class defined above" because quantification over reals (proper-classes) or intervals of reals (super-classes) is not permitted in my logic NAFL. I have not yet formulated in detail how I am going to do various aspects of real analysis by this method. As of now I can no longer continue this work because my bosses at IBM have given me other work and told me that this logic stuff should be not be done on official time. Frankly, I am now struggling for survival here.