Re: Calculus XOR Probability

W. Dale Hall said:
Tony Orlow wrote:

... stuff deleted ...

(I know, I've responded elsewhere, but noted a couple of points
that I had missed. Apologies for whatever redundancy has been
generated, and the concomitant acceleration of the heat death
of the universe.)


BTW, it seems like you *still* haven't solved the problem of

"everything is divisible by x"

for *every* natural number x coprime to 10. There's no getting around
this for your version of the integers. I've already shown you how to
do that division, so you can't claim it ain't so.

Shouldn't that be an important issue? After all, what use are your
numbers if you can't even do arithmetic? If your numbers are of no
use (i.e., literally "useless"), then aren't they worthless?


Hi Dale. Been a long time. So long, it appears, that you don't seem to recall
the lengthy response I gave to your "everything is divisible by 7" argument,
where I fully analyzed what you were doing, and confirmed that, with the 10-
adics, indeed ...110, ...111, ...112, etc are apparently divisible by 7. I
explained why this was so, pointed out that in fact ...1115 is NOT divisible by
7, since there is no multiple of 7 consisting of a string of 1's ending in a 5
as is true for the other 9 digits, and finally, reminded you that my infinite
number system is distinctly different from the adics. They're not "left-
infinite", but "center-infinite". I then applied the division by 7 to my unit
infinity, 1:000...000, and found that, with a six-digit repeating sequence in
decimal 1/7, the remainder of dividing decimal infinity by 7 could be any of
six different values, from 1 through 6, and could not be 0, which makes sense,
since no power of 10 could be evenly divisible by 7. So, I showed that my
numbers work there the adics don't.

But why do you say that no power of 10 could be evenly divisible by 7?

First of all, because 10 and 7 are coprime, but you know that. Any system which
claims a power of 10 is divisible by 7 has got problems. I would have abandoned
any such system as useful.


0:0000....010 = 0:000...28571428571430 * 7

The center-infinite string has its left string all zeros, and its right
string equal to the periodic sequence 285714 terminated on the right by
30.

That's not a T-riffic. There is no specification of where the 0's become the
repeating pattern. I don't know how you can even suggest something like 10=7*
28571428571430. Do you have a straight face? Is 7*2857142857143=1? I can't even
tell what operations you think you're performing on your number system.


As for your "unit infinity",

1.000...000,

isn't it equal to

...2857142857143.000...000 * 7 ?

What on Earth makes you say that? 7*3 is 21, so whatever answer you get would
end in a 1. After that it produces 0's, so you might consider that to be equal
to ...0001, I suppose, but saying 7 times an infinite number equals 1 makes no
sense. It looks like the 1/7 repeating pattern, but you've rounded the
2.8571452... up to three? No, this is not right.

1:000...000/7 is one of the following:

0:142857....142857.142857...142857 with remainder of 1
0:142857....1428571.428571...428571 with remainder of 3
0:142857....14285714.285714...285714 with remainder of 2
0:142857....142857142.857142...857142 with remainder of 6
0:142857....1428571428.571428...571428 with remainder of 4
0:142857....14285714285.714285...714285 with remainder of 5

Since the repeating pattern continues forever, it does not terminate at the
zero point, and therefore does not evenly divide any power of 10. Where the
zero point falls in the repeating pattern determines the finite remainder, but
that can't be determined in such an infinite division.



(Here, the string to the left of the period extends periodically,
endlessly repeating the string 285714)

Or does the string to the left of the period not admit (my version
of) infinite strings of digits? Any time you admit infinite strings
of digits, the same operation can be applied, without fail. The
presence of a left-terminating infinite string of digits doesn't
save your arithmetic, as another of this spate of redundant posts
of mine has shown. The jist of my argument is that the left portion
of the string can be divided separately, just as one can produce
the periodic decimal expansion of a standard fraction. The fact that
the left and right strings (on the separate sides of the ellipsis)
do not interact allows one to divide both strings separately: there
is no digit position that admits a carry operation from the right
string to the left string.

Only finite-length strings of digits can fail to admit such division.

Oh, criminy! If you want to critique the system, at least get it right. The
decription of it has been refined somewhat since we last discussed it, so let
me review the T-riffic rules:

There is a digital point to the right of the digit representing the 0 power of
the base, called the zero point.

There are other digital points called limit points, which may be infinitely to
the left or right of the zero point, which are denoted by a finite number of a
formula on N, the unit infinity.

Around each digital point is a countable neighborhood within which is defined
an explicit finite string.

Between the explicit finite strings in each countable neighborhood extend
finite or infinite strings of repeating digits.

Within the neighborhood of the leftmost limit point must begin a left-infinite
string of 0's for positive numbers.

The binary version can work just like 2's complement, allowing for left-
infinite strings of 1's for negative numbers, if you want. Decimal versions can
use a 10's complement system for negatives, but I think this may lead to some
issues, so I generally use the binary T-riffics, but for positives, there is no
issue. There must eb an explicit leftmost digit, which may be in an infinite
position.


So, you can claim that I never solved your "problem", that arithmetic doesn't
work with the T-riffic numbers, and that my ideas are worthless, but that's a
bunch of baloney, and if you have any memory of the response which I describe
and a shred of integrity, then you'll admit that I exposed your tricky
maneuver, which I don't think you invented, and that the objection was vacuous
and irrelevant. Okay?


I looked up a response that appears to be the one you are referring to;
it was dated December 1, 2005. My latest contribution to that thread had
been on August 19, and while I followed the thread for about a month
thereafter, I didn't hang around for three and a half months while you
got your act together. I found the thread "Infinity" to be unbearably
tedious and contentious, and tired of it. If it is my fault for having
too short of an attention span, then I can accept that.

It's no biggie. You don't have to read my posts. But, if you want to criticize,
you should know what you're talking about.


Your language (in referring to my "tricky maneuver") suggests that
my suggestion (that everything is divisible by 7) is not to be taken
at face value. You may accept or deny anything you please, that is
of no concern to me. However, if you're serious that you have somehow
defined a structure that represents integer arithmetic, you'll need
to show how arithmetic works, and how it agrees with what the rest
of the world sees as arithmetic. I don't see how you've done that.

I don't get the feeling you've looked very hard, since what you are doing above
bears little or no relation to the T-riffics.


My point is this: your number system doesn't admit arithmetic. That's
all. If you wish to claim that it is an adequate model of the naturals,
then that's fine, but you'll have to admit that arithmetic doesn't
really exist.

My point is this: It does, and you think it doesn't because you don't know the
system. If you follow the rules of the system, which are different from the
adics, the it works, no matter how many times you calim it doesn't.


Maybe you disagree. Then show me that I'm wrong. In my adult life, I
have never minded being shown to be wrong when I am, and will not have
any problem with it now. I don't have any ego involvement here.

You got the goods? Then go to it, youngster.

Uh huh. Try responding to the stuff I wrote above and see what questions you
have after a reminder of what it is we're talking about. Thanks.


Dale.


--
Smiles,

Tony
.