Rotwang wrote: > On 15/03/2012 02:50, DavidW wrote: >> Rotwang wrote: >>> On 15/03/2012 02:24, DavidW wrote: >>> Were you to actually try to answer the question, rather than coming >>> up with decreasingly inventive ways to avoid answering the >>> question, you would find out why it matters and actually learn >>> something. But if you'd prefer to continue desperately squirming >>> for the entertainment of people who already know some maths then >>> that's fine too. >> >> There's nothing "inventive" about that answer. It's easy to >> construct an infinite sum with a limit of, say, 2. The sequence >> won't sum to 2, but you can get arbitrarily close to 2 and you can >> say that's its limit is 2. Similarly, another sequence might have a >> limit of e. > > It might. But knowing that a sequence /might/ have a limit of e is not > all that useful as far as mathematics is concerned. I'm asking you to > give a definition of e and /prove/ that a number satisfying said > definition exists.
I meant that there might be such a sequence, not that a given sequence might have that limit.
> Look, I'll make it a bit easier for you: a series that works is given > by a_n = 1/n!. I know how to prove that that series has a sum, in the > sense of the definition I've given, and that sum is one of several > equivalent definitions of e. If you'd like me to I will show you the > proof. But it uses infinite sets.
If you like, but I still question the relevance. You seem to be arguing that because you can do these fancy and interesting things with infinite sets it's okay to just skate over any logical objections or contradictions within them.
>> Look, all I'm saying that it's a contradiction to speak of "all" >> natural numbers, since there's no such thing, and treating infinite >> sequences as though they terminate when they don't. It's a very >> simple proposition, and if it has consequences for e or anything >> else, then so be it. > > You've also said that the axioms given at > http://2000clicks.com/mathhelp/BasicArithmetic.aspx are the > foundations on which all mathematics stand, and that you don't have a > problem with calculus.
Not so far, but I haven't looked very hard.
> But calculus makes liberal use of the > existence of the number e, and those axioms do not suffice to prove > the existence of e, unless the sets to which the completeness axiom > refers can be taken to be infinite. If you think otherwise, then show > me.
I don't know if I can or not. I also question the relevance of this.