123.3 + 0.1 is 123.3999999999 ?

[Moshe Zadka]

On Wed, 04 Jun 2003, Erik Max Francis <max at alcyone.com> wrote:

> There are all sorts of alternative definitions of the reals which have
> differing properties than the reals we've all come to know and love. 
> They usually fall under the general category of "non-standard analysis."

In non-standard analysis, 0.9999..... has *no* meaning if you just translate
the usual definition of limit from the real numbers. This shouldn't be
surprising, as limits are not a first-order property. In general, since
being a least upper bound is not first-order property, there are no
least upper bounds in non-standard analysis. And again, not surprising --
it is easy to prove that any ordered field with least upper bounds is
the standard reals. Only when we become myopic enough for first-order
sentences only, can the non-standard reals simulate the reals.

(Oh, of course, if you translate the definition of limits blindly
*enough*, that is to allow the sequence to range over the non-standard
integers too, then 0.999....=1 again. Although the *name* 0.99999...
is misleading, since it looks like it only ranges over the standard
integers).
-- 
Moshe Zadka -- http://moshez.org/
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