Re: Well Ordering the Reals

Virgil said:
In article <MPG.1e5bbe88fbf4af7f98aa3d@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

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.

TO is, in effect, declaring that his number system is not subject to
logic. Non-logical systems, such as his, are also non-mathematical, as
mathematics conforms to logic.


I am saying it is not subject to the standard set theoretical axioms regarding
what it calls sequences. Your claim that a sequence can't be infinite is bogus,
and a measly attempt to subject the only rational expression of infinite
numbers devised to date, to your petty constructions. :) See? I can be pompous
too. Pretty good, huh?


They are a simple extension of the digital
number system allowing for values of infinite ratio within the set.

But they extend to properties of sets beyond the bounds of any known set
theory and in direct conflict with any known set therory. So that,
absent an axiom system consistent with them, they are of no more
subsstance than castles in air.

What properties of the set is "beyond the bounds of any known set"? I already
showed you a sequential ordering of the reals, even if you don't believe it, so
there you have an uncountable sequence. We are not amused with your finite
infinities, snicker as we may.


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).

TO has been in silly moods every day he has posted here. Does he have
any other kind?

Oh sure. Sometimes I am quite serious about these things, and other times it
rather amuses me. Do you really think I am always silly, or just plain moronic
and crankish sometimes? :)


It is not my job to account for your misconceptions regarding
infinite sequences.

That's our line!

Sue me!



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.

One can define all sorts of things that do not exist, like four cornered
triangles and TO-numbers. One needs to see PROOF that the defined
objects actually exist before one need regard them as anything but
fantasy.

I suppose, to be mathemtically rigorous. And yet, if someone said, "why don't
you run the bus's exhaust pipe up the back, so it spews out the top, and not
into people's faces," the equivalent of your position would ebe to demand a
full schematic of everything in the car before even considering if the idea
made sense.


Therefore, the T-riffics are
an empirical counterexample to your "proof" that no such sequence can
"exist".

No more empirical than four cornered triangles!

Well of course trianguloids have four vertices in three dimensions. You would
call this a tetrahedron.


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.

Still no more empirical thatn 4 cornered triangles.

Are you honestly saying that you can't see that the normal finite system is the
T-riffic system with a single limit point at the 0 bit? One might wonder at you
not having fallen down a well or something, but perhaps you're tied to your
computer.



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.

I am only describing TO's position. It is his fault that he is in it.


Oh, I feel quite lucky to be able to discuss this stuff with people all over
the world. Life's a miracle, even with all the ***. Have a nice ***, uh, I
mean, life!
--
Smiles,

Tony
.