Re: .9 repeating

lwalke3@xxxxxxxxx writes:

I don't yet believe that there can be no rigorous
theory in which there are infinitesimals with digits
or a smallest infinitesimal.

Well, of course there can, depending on what you mean. I gave a
rather dull suggestion earlier. Let me repeat it here, hopefully a
touch cleaner

Rather than give an axiomatic theory, I'm going to describe a
structure. From this structure, I reckon one could construct a theory
easily enough.

My structure is the set R' of all functions

(w + 1) -> {0,...,9}

There's a natural injection [0,1) -> R' defined as follows. Let x in
[0,1) and let

(x1)(x2)(x3)...

denote the decimal expansion of x. If x has two decimal expansions,
one ending in 0000... and the other in 999..., choose the former.
We map x to R' via

x |-> <x1,x2,x3,...,0>

(that is, to the function f such that f(i) = xi and f(w) = 0).

Okay, so R' is an extension of [0,1), but now we need a *different*
map interpreting digit strings as elements of R'. Note: so far, we
embedded actual real numbers into R' using their decimal expansions,
but *now* we are mapping sequences of {0,...,9} into [0,1), so this is
a somewhat *different* matter.

The map is nonetheless very similar. We map a sequence

<x1,x2,x3,...> |-> <x1,x2,x3,...,z>

where z is defined as follows: If the decimal expansion
(x1)(x2)(x3)... eventually stabilizes to a digit k, then pick z = k.
That is, if there is an n such that for all m > n xm = k, then z = k.
Otherwise, let z = 0.

where z is defined just as above.

Notice, now, that we have interpretations of the set {0,...,9}^w
(let's write it as X^w) into both R' and [0,1).[1] We also have an
embedding from [0,1) into R', like so.

X^w -> R'
| ^
| /
| /
v /
[0,1)

This diagram does *not* commute. The sequence <4,9,9,9,9,9,...> maps
to 0.5 in [0,1) and 0.5 maps to <5,0,0,...,0> in R', but
<4,9,9,9,9,9,...> maps to <4,9,9,9,9,9,...,9> in R'. Similarly,
<1,1,1,...> maps to 0.111... in [0,1) which maps to <1,1,1,...,0> in
R', but <1,1,1,...> maps directly to <1,1,1,...,1> in R'.

Okay, now intuitively, the elements <0,0,0,...,n> are the
infinitesimals in R' (where n > 0). The mapping X^w -> R' is our
interpretation of decimal notations. Under this interpretation

(1) 0.999... is the greatest element in R' and so is intuitively below
1 (since we interpret R' as an extension of [0,1), but not an
extension of [0,1]).

(2) The element <0,0,0,...,1> (which we might write as 0.000...1) is
the smallest positive element.

Now, could we define addition and so forth? Perhaps. It's a little
hard to know what to do when the w'th position has a carry, but maybe
there's something sensible to be done (I kinda doubt it).

If there is no sensible addition, then this is clearly a dead end. If
there is a sensible addition, this is still almost certainly a dead
end. It is just unlikely in the extreme that anything approaching
real analysis could be done on this toy model.

This is why I think you don't quite express your project well. It's
not enough to find a theory or structure making two or three or n
pronouncements true. The theory has to be useful, since it is
supposed to be analogous to the remarkably useful theory of real
numbers.

Good luck with that.

Footnotes:
[1] Not really, since the sequence <9,9,9,9,...> does not map into
[0,1), but we can speak of X^w \ {<9,9,9,9,...>} if precision is
necessary.


--
"I've noticed [...] I routinely have been putting up flawed equations
with my surrogate factoring work. My take on it is that I have some
deep fear that the work is too dangerous and am sabotaging myself."
-- James S. Harris
.

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