On Mon, 3 Mar 2014 09:23:39 -0800 (PST), John Gabriel <firstname.lastname@example.org> wrote:
>On Monday, 3 March 2014 19:14:16 UTC+2, dull...@sprynet.com wrote: > >> Of course I know what the general term is. I know a lot of things. The >> question is how do we know what the general term is, _from_ the >> derivation of the first three terms you showed me. > >How do you know about the general term if not from the first few terms? There is no other way to know it. :-) > >> I asked how you knew what the sine series was. You showed me that >> derivation of the first three terms. How do _you_ know what the >> general term is? > >The first 3 terms is how I know the general term.
What??? That's not a proof. Not even close.
> >> Oh for heaven's sake. The things you say on that page have essentially >> nothing to do with Dedekind cuts. > >That is absolutely FALSE. The information on that page describes exactly what Dedekind was trying to do. His cuts are absolute rubbish. Until I came along, no one was able to write the "things" you read on that page. > >> That's putting it too strongly - it could be that those things are useful as >> a way to understand Dedekind cuts (could be, not that I really think so). > >Um, no. Those things are useful to expose Dedekind for the idiot he was. :-) > >> But they have nothing to do with the actual _definition_ of Dedekind >> cuts - the defiinition does not involve limits. > >You are wrong! Limits are very much involved. The definition does not include a reference to limits, but a D.Cut is in fact meaningless without limits.
Look. Here's a simple example. Say R is the set of all rationals r with r > 0 and r^2 > 2. Say L is the set of all rationals r with r < 0 or r^2 < 2. Then the pair (L, R) is a Dedekind cut. If we're taking Dedekind cuts as our definition of the reals then (L,R) _is_ a real number.
Read that again. The pair (L,R) _is_ a real number. Literally.
Nothing there about limits. Nothing there about some number that trapped between L and R. If there _were_ something in the definition about a number trapped between L and R then you'd be perfectly justiified in being skeptical about why we should believe such a number exiists.
But there's no such mystery involved. (L, R) is a real number. Period.
> >> But you've inspired me. I see a very simple proof that John Gabriel does not exist. > >Meh, I have no idea what you mean by that, and I don't think I care either. ;-)