On 13/03/2012 01:16, DavidW wrote: > Peter Moylan wrote: >> [...] >> >> I'm in the Cantor camp myself. If you can't conceive of uncountable >> infinities then measure theory has to be re-done from the bottom up, >> and things like integration become conceptually harder. Not to mention >> concepts like continuity, limits, and so on. > > I don't have a problem with calculus, limits, etc., or any sort of convergence.
You would do, if you actually understood them.
Exercise: using what you claim are "the axioms of mathematics" (at http://2000clicks.com/mathhelp/BasicArithmetic.aspx), give a definition of the number e > 0 and show that it has the property that d/dx (e^x) = e^x. This will involve giving a definition of a^b for arbitrary real b and positive a, which has the usual properties which are required of exponentiation.
Hint: the usual definition of e is as the least upper bound of the set of partial sums of the form sum_{i = 0}^n 1/n!, n in N. But since that set is infinite, and therefore not a set (according to you), the completeness axiom may not be used to show that such a least upper bound exists.
