Re: .9 repeating

lwalke3@xxxxxxxxx writes:

Indeed, this entire thread has been devoted to seeing
whether or not there can be a theory in which one can:

1) Represent positive infinitesimals with digits.
2) Find a smallest positive infinitesimal.

Rigorous theories have been found in which either 1) or
2) are true, but one in which both is true is lacking.

I'm not sure it's so hard to do both, you know, if you don't require
that *every* "infinitesimal" is represented.

Let's let R' be the set

{f | f: (w + 1) -> {0,...,9}} u {1.000...0}

That is, let R' be the set consisting of "(w + 1) digit sequences"
(this is, according to some, an abuse of the word sequence, but let's
continue).

There's a natural ordering on R', namely the pointwise ordering.

0.000...0 < 0.000...1 < 0.000...9 and
0.999...0 < 0.999...4 < 0.999...9

There's an order-preserving embedding [0,1) -> R', defined as follows:

We map x in [0,1) to the sequence

(x1)(x2)(x3)...0,

where xi is the i'th digit of x, with the proviso that if x has two
different decimal expansions (one ending in all 9s and the other in
all 0s), we use the expansion ending in all 0s.

Now we make an arbitrary choice: we interpret infinite digit sequences
(note: here I'm interpreting syntactic objects, not elements of R)

(x1)(x2)(x3)... |-> (x1)(x2)(x3)...0 unless the sequence ends in
all 9s
(x1)(x2)(x3)... |-> (x1)(x2)(x3)...9 if it ends in all 9s.

Now according to this, 0.999... is the largest number in R', analogous
to the largest element in [0,1). There is a smallest positive number,
which is 0.000...1, but I haven't assigned a digit sequence to this
number, because I interpreted only omega sequences. If I extend this
interpretation (like Mitch?) to the "sequence" 0.000...1, then this
guy has a name.

This structure satisfies Mitch's desiderata, more or less.

But it's also completely uninteresting. I don't see why anyone would
want to study it at all. I see no reason to believe that analysis
done in this structure would be interesting or fruitful and I don't
see that this structure matches any of my intuitions about numbers
better than classical analysis.

As far as interesting developments, of course, I could be wrong.
Maybe there are some keen features of a structure something like this,
but I certainly wouldn't invest my time on the hope that such features
exist.

I'm sure that this post is chock-full of errors, since I tossed it off
as quickly as possible. My point is merely that satisfying (1) and
(2) together are neither enough to vindicate MR as a solid
mathematical thinker *nor* enough to justify the claim that theories
in which (1) and (2) are true are worth pursuing.
--
Jesse F. Hughes
"Besides, discoverers are too proud to kiss ***. Indiana Jones would
never kiss some academic's ass to get published, and neither will I."
--James Harris
.