Re: Well Ordering the Reals
- From: Tony Orlow <aeo6@xxxxxxxxxxx>
- Date: Tue, 14 Feb 2006 10:17:20 -0500
Virgil said:
In article <MPG.1e5a76a391d3f51c98aa33@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
David R Tribble said:
Virgil wrote:
TO-numbers cannot support a standard arithmetic as described.
Tony Orlow wrote:
The arithmetic isn't exactly standard. It's somewhat restricted. But,
there is
arithmetic possible with these numbers, and they are an improvement, I
think,
over the adics.
That could be your first useful proof. Take an arithmetic problem that
is difficult or impossible using p-adics and show how easier it is
using T-riffic numbers.
Well, we could talk about how the 10-adics ...111, ...112, and ...113 are all
apparently evenly divisible by 7, which makes no sense, whereas 1:000...000
when divided by 7 has 6 possible remainders, none of them 0, which makes
perfect sense. Or, we could talk about how the infinite sum of the divergent
infinite series 1+2+3+4... can be expressed precisely as
0:100...000:100...000,
with limit points as 2log2(N) and log2(N). We could talk about how the T-
riffics handle negative values as well as positive, and how addition works
between them as it does for normal n-complement numbers. What other problems
are there with the adics? The T-riffics probably solve all of them. :D
But TO-numbers cannot solve the problem of how they can exist when all
standard set theories say they cannot.
The T-riffics are not subject to your puny standard theories and their bogus
theorems. They are a simple extension of the digital number system allowing for
values of infinite ratio within the set. They scoff at your aleph_0, your
cutest little infinity, and countability as a criterion for anything. The T-
riffics are immune to your proofs. Haha (I am in a silly mood today).
It is not my job to account for your misconceptions regarding infinite
sequences.
TO-numbers require a sequentially ordered uncountable set to index it
digits, but all standard set theories, with their standard inductive
axioms, allow simple proofs that sequentially ordered sets are, at most,
countably infinite.
And yet, it's funny how declaring bits at infinite positions in the string
hasn't led to any obvious contradictions. Given the repeating strings defining
the infinite distances between limit points, an uncountably infinite string is
defined. Therefore, the T-riffics are an empirical counterexample to your
"proof" that no such sequence can "exist".
Now, as far as being able to specify any possible exact sequence, this can only
be done over a countable domain. So, given only the zero point, we have only
finite numbers that can all be specified. When you add limit points at infinite
positions, you add new countable neighborhoods in what is really, ultimately,
an uncountably long string of bits, and this doesn't break anything concerning
the finite numbers. They are the subset of the T-riffics that only contain a
limit point at 0, normally, or in general, only contain limit points at finite
positions, and the question of infinite repeating strings between limit points
is moot.
SO that absent a TO axiom set for his TO numbers which somehow allows
uncountably infinite sequentially ordered sets, TO is SOL.
Don't be such a potty mouth. You're setting a bad example for the kids. ;)
--
Smiles,
Tony
.
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