From: Lee Corbin (lcorbin@tsoft.com)
Date: Sat Oct 12 2002 - 11:41:22 MDT
Dan writes
> >[Ross wrote]
> >> That leads into indefiniteness, like, what is
> >> the smallest real number, (iota), whatnot.
> >
> > There isn't a smallest real number. Or there is.
>
> I don't know what Ross is talking about here. There
> isn't a smallest real number in standard analysis. The
> reals are dense (as are the rationals), meaning as Ross
> seems to get, between any two of them, there's another.
>
> In nonstandard analysis, there is an infinitesimal, but
> these are superadded onto the reals. I'm not familiar
> enough with nonstandard analysis (hyperreals) to know
> if the idea is more than a parlor trick.:)
As a devout mathematical platonist, I believe in the
infinitesimals, after the revelations of St. Robinson.
However, I believe that there cannot be a smallest
non-standard entity here either, and the reason is
the "game" that seems to have become the ultimate
arbiter for the last century or so. Were someone
to announce a smallest real or hyperreal, we would
retort that we could find a yet smaller one, and
indeed prove the point by exhibiting one exactly
based upon his contender.
It is curious how mathematical truth appears often to
depend upon who has scored the most debating points,
or at least who gets in the last word! ;-)
Lee
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