(February 2, 2012 at 8:57 am)Categories+Sheaves Wrote: I'mma drop some links
See: Wikipeida or [url=http://www.math.princeton.edu/~nelson/books/1.pdf]Ed Nelson's introduction to nonstandard analysis[/i] for systems of analysis where this sort of thing need not happen.
There systems do treat infinitesimals as genuine, nonzero quantities. Here we have that .999... ~= 1 i.e. they are infinitely close, but we do not insist that .999... = 1.
Now, I'm a big fan of the standard real numbers, and I strongly prefer using them to the other systems out there. But the other systems are still out there. We do get this equivalence of .999... and 1 once we accept all that business with cauchy sequences and epsilon-delta. But definitely not before then.
Well, of course I assume we're talking about the standard presentation of the real numbers, which are an Archimedean field.
(But I suspect that even in the hyperreals, .999... = 1.)
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
