Paul Lutus wrote: > "Mike Oliver" <oliver@math.ucla.edu> wrote in message > news://3C07175E.6985DD03@math.ucla.edu... > > Lutus, you're being an ass. Solipsism doesn't enter into it. > > It didn't until you drifted into it. Then it became the issue. You have > chosen to ignore any commonly accepted definitions appropriate to the > original context. That is how solipsism is defined.
Find me these "commonly accepted definitions" in some respectable source. Show me in Bourbaki where it says a "number system" has to satisfy additive cancellation. Show me in Lang, show me in Hungerford, show me in Folland, in Rudin, in *any* decent graduate-level algebra or analysis text. You can't, because no one writing at that level would say such a silly thing.
> And, regardless of the multitude of special-case definitions, consider the > context. A student asks: > > > Is zero an approximation? > > An approximation of zero, where zero is actually 1/infinity? > > For someone who posts an inquiry like this, my reply is the appropriate one. > Zero is not "actually 1/infinity."
Well, in the extended complex numbers, it actually is. And it is very possible that the extended complex numbers are the natural structure to capture the intuitions he was trying to get at, even if he expressed them poorly.
They *are* an extremely natural structure, you know. You just take C, one-point-compactify it, and then extend the basic operations to the maximum domains they can have subject to continuity. This can be done in a unique manner, and it's both a natural and a useful thing to do.
Their algebraic properties are not super-mega-friendly. Tough titties. It's a luxury to be able to specify the algebraic properties you want your structure to have. Lots of times, the structure you want to consider is fixed, and its algebraic properties are what they are.
