So here goes a nice lengthy response...

>>>The idea that they swallowed is the definition of the limit. That definition is reasonable; it works; and as you know, is a fundamental part of the definitions of the derivative and the integral. No problem.>>The reciprocal of 0 is undefined. There is no way of defining it that does not lead to contradictions.>No. 'Undefined' does not mean 'infinite'. Come to think of it, though, that's not a bad idea. We should all agree that the term 'infinity' is undefined, and thereby disallow any use of the term. I would vote for that.>>Mark: But perhaps we can define "zero" within the context of George Cantor's transfinite numbers...(??)

Paul: I suppose you could do that. If you did, I am quite sure you would produce a number of antinomies, just as Cantor's definitions themselves did.>Mark: 1 - .9999999 = "almost zero but not just quite".>>What you are doing is confusing the notation

.99999

for the notation

.99999...>>The axiom of choice says something like, if you can come up with some algorithm for producing an element with ordinal number n+1, given the existence of an element with ordinal number n, then you can assume that such an element exists for any ordinal m>n. (Or something like that).

That is where I holler "W A I T a minute! You can't do that!" But all mathematicians do it anyway.>>As for physics, it appears at this point that everything in nature has a grainy texture if you get down small enough. Don't you think a grainy math might be better suited to physics than the smooth math with its troublesome singularities (infinities)? I rest my case.