Adopey.1985

Adopey.1985
net.math
utcsrgv!utzoo!decvax!duke!unc!dopey.bts
Fri Mar 19 10:46:33 1982
0.9999... infinity

Then, the real numbers are the equivalence classes of these
rational sequences. The rational numbers we started with
are "embedded" in the reals by identifying them with con-
stant sequences. Hence, the real number 1/2 is the
equivalence class of the rational sequence
1/2,1/2,1/2,1/2,...
To construct a simple "non-standard" number system,
start with sequences of reals and an ultrafilter U on the
natural numbers, the indices for our sequences. (Here's all
you need to know about the ultrafilter we'll use. It's a
collection of sets of indices. Every set of indices with
finite complement is in U. Every set of indices is in U or
its complement is in U. And, if a set of indices is in U,
then any super-set of indices of that set is in U as well.)
Now we say that two sequences s and t are equivalent, s ~ t,
if
{ n : s(n) = t(n) } is in U.

Now, as you'd expect, the non-standard reals are just the
equivalence classes of the sequences of reals. The standard
real numbers are embedded in the non-standard universe by
identifying them with constant sequences. For instance, the
real number pi is associated with the equivalence class of
the sequence pi,pi,pi,pi,... , a typical infinitesimal might
be the equivalence class of the sequence

0.1, 0.01, 0.001, 0.0001, 0.00001, ...

and a typical infinite element might be the equivalence
class of the sequence

1, 10, 100, 1000, 10000, 100000, ...

Finally-- and this is the first deviation from what
you'll find in a text on non-standard analysis-- let's agree
on the following interpretation of non-terminating decimal
fractions. If x is a non-terminating decimal fraction, let
x(n) be the n-th symbol of x, read from left to right.
Associate x with the equivalence class of the sequence sx,
where

sx(n) = x(1)x(2) ... x(n)

This means that 0.9999... will be associated with the
equivalence class of the sequence

0, 0., 0.9, 0.99, 0.999, 0.9999, ...

There's one more technical detail, then I'll get back
to 0.9999... and 1. In general, a formula about two ele-
ments of the non-standard universe is true if, when you take
a sequence from each equivalence class the set of indices
for which the reals in the sequences satisfy the formula is
in U. Let s be the sequence of associated with 0.9999...
and t the constant sequence 1. Then 0.9999...
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Bruce Jones, Henry Spencer, David Wiseman.