Re: The infintely small number b ... B = aleph_-3 ?

On Sun, 18 Nov 2007 17:38:20 EST, tommy1729 <tommy1729@xxxxxxxxx>
wrote:

david wrote:

On Sat, 17 Nov 2007 09:46:21 -0800 (PST), Venkat
Reddy
<vreddyp@xxxxxxxxx> wrote:

On Nov 17, 10:05 pm, David C. Ullrich
<ullr...@xxxxxxxxxxxxxxxx>
wrote:
On Fri, 16 Nov 2007 20:45:47 -0800 (PST), Venkat
Reddy





<vred...@xxxxxxxxx> wrote:
On Nov 17, 8:55 am, William Hughes
<wpihug...@xxxxxxxxxxx> wrote:
On Nov 16, 10:44 pm, Venkat Reddy
<vred...@xxxxxxxxx> wrote:

On Nov 17, 8:32 am, William Hughes
<wpihug...@xxxxxxxxxxx> wrote:

On Nov 16, 10:21 pm, Venkat Reddy
<vred...@xxxxxxxxx> wrote:

Following is the definition and
properties of the infinitely small
number b which I called sookshma number
in my recent posts. This may
also be called infinitesimal, but I'm not
sure if the formal
definition and properties of
infinitesimal are the same. I'm using b
as the symbol for this, simply because it
was used in the recent
discussions.

I knew this definition is neither
rigorous nor complete, but I think
it has the essential ingredients to make
it more formal.

The infinitely small number b
-----------------------------

b is the number that represents smallest
extent.

b is not equal to zero because zero
represents non-existence of
extent, while b represents existence of
extent.

b is not equal to b in arithmetic sense
(addition, subtraction).

b is equal to b in geometric sense: b*1 =
b; This simply means that b
exists

Operations with zero (non-existence of
extent)
b + 0 = b
b - 0 = b
b * 0 = 0
(b / 0) is undefined and not to be taken
as inf.

Operations with itself
b + b = b
b * b = b
b / b = 1
(b - b) is undefined.

Operations with n where b<n<inf:
n * b = b
n / b = inf. (indicates that n/0 is
undefined and can not be taken as
inf.)
n + b = n + b
n - b = n - b

We can't precipitate the last two
arithmetic operations (addition,
subtraction) because it requires
comparison of b with b itself in
arithmetic sense which is undefined.

- venkat

The above is not a definition. It is a
wish list
of properties that you would like something
called
b to have. You want b to be a "number",
but clearly
b is not a real number or an extended real
number.
You can't define b to be equal to 1/inf
unless
you define what you mean by 1/inf.

It is actually n/inf, not 1/inf.

This is even sillier. You don't know what
1/inf is
so to get around this you use n/inf!?!.

The n/inf is the extent of a piece
when a piece of extent n is cut into inf
pieces.
inf is the
uncountably large number, which is what you
get when you keep counting
things for ever.

Actually this infinity is countable infinity.
Uncountable infinity
is something different.

Fine. I have no problem in calling it as
countable infinity, if that
works for you.

You're concentrating too much on unimportant
details and ignoring
the main point, which WH expressed very well:

I haven't ignored WH's comment. There was no
question to be answered
there. WH says b is not real number - I have no
issues.


"The above is not a definition. It is a wish list
of properties that you would like something called
b to have."

The description of b was not rigorous enough to call
it a definition -
I've acknowledged it in the description itself.


Whatever your b is, it certainly cannot be a real
number.

Fine.

You need to say what sort of thing b actually
_is_,
and prove that there _is_ something with those
properties.

This, I guess, is _your_ demand. The description I
gave answers this
question to some extent. I'll try explaining it
further. The basic
assumption here is: Zero extents can't build up to a
finite value even
if multiplied by infinity times. So, n/0 is not
infinity and n/inf is
not zero for 0<n<inf; they are just undefined values
due to special
nature and role of zero. This situation requires a
smallest magnitude
to serve as counterpart of inf on the smaller side
of the scale. In my
view, this smallest number has very similar
properties and as much
existence as the largest number, in this case a
countable infinity.

None of that says anything whatever about the
question of what b
actually _is_.

Evidently b is the smallest positive number. There
_is_ no
smallest positive _real_ number. So what _is_ b?

Just saying that some situation "demands" something
does
not prove that that thing exists.

- venkat




- venkat

************************

David C. Ullrich- Hide quoted text -

- Show quoted text -


************************

David C. Ullrich

amazingly , i agree with david.

b only exists in the head of the OP.

at best it could be used as some kind of code.

however note that b does remind me of some bogus aleph-3 math.

b =/= aleph_0 =/= aleph_1

b-b = not defined , just as aleph-3 substraction.

n*b = b ...just as n*aleph_x = aleph_x

so i agree on the critisism about b.

but i strongly note the simularities with aleph_3....

so when someone defines something similar to aleph_3 math but called it b, it is suddenly noticed invalid.

but if someone uses the same properties and types aleph_3 it is good and consistant ?

Yes. The big difference is that we _can_ say exactly what aleph_3
_is_.

(aleph_0 is the set of all finite ordinals. aleph_1 is the set
of all ordinals of cardinality less than or equal to aleph_0,
which is to say the set of all countable ordinals. aleph_2
is the set of all ordinals of cardinality less than or equal
to aleph_1, and finally aleph_3 is the set of all ordinals
of cardinality less than or equal to aleph_2.)

or when someone uses the b-counterarguments and types aleph_3 , he is wrong , despite these arguments now used by the ones supporting aleph_3 ???

feel the power of hypocritical set theorems !!!

oh , i dont blame david much , he is a child of his time ...

You'd really look less ridiculous if you restricted your
comments to things you understood.

regards
tommy1729


************************

David C. Ullrich
.

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