Rotwang wrote: > On 13/03/2012 01:16, DavidW wrote: >> 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.
Does this come down to a quibble over what a set is? It's a converging infinite series. The further you go the more accurate e is. I don't see any problem with this.
