Re: The infintely small number b

On Nov 21, 2:38 pm, mike3 <mike4...@xxxxxxxxx> wrote:
On Nov 18, 5:18 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 18, 9:56 am, mike3 <mike4...@xxxxxxxxx> wrote:
<snip>
Why do you need bee - bee to "bee" (heh) undefined?
Why not bee - bee equal zero?

It is not equal to zero because of the first rule: b is not equal to
itself. b is not a single number but the "smallest" number. So it is a
quality of the things (quoting Buddha Thu referring Gauss).

But why cannot b equal itself? You say because it is the
"smallest" number, but how does that imply b != b?

Yes. The "smallest" number can't equal itself in the same sense as the
"largest" number can't equal itself. As you were going to say,
equality operation does not make much sense here.

In your
number system, then, "equality" does not exist, or is no
longer an equivalence relation as the reflexive law fails.

Equality still exists for finite numbers which are collections of
infinite number of b. The reflexive law fails only at very small scale
of finite number of b, and hence b != b.

- venkat








Also, why don't you like the surreal numbers? They
do what you want, make infinitesimals. And they make
lots more than your bee-number (or whatever that
other silly name was you gave it), and the best thing
is they've got a rigorous definition, which yours
doesn't.

Yes, I'm not formally trained in mathematics, so please pardon the
lack of rigor in my writings. Regarding surreals, I have just now
taken a look at them on wikipedia. Conway's infinitesimal allows b = b
and in general the operations go too far making the infinitesimal look
like a single definite number, but not a symmetric counterpart of
infinity in geometric sense (or multiplicative sense). So I think b
does not adapt to the definition and properties of surreal
infinitesimal. I haven't looked at hyperreals - will take a look and
see if b is superfluous.

- venkat

What do you mean that it isn't a "symmetric counterpart" of infinity?
The infinitesimal surreals are "smaller" than every other real, just
as the infinite ones are "larger". There does not seem to be a reason
why an infinitesimal cannot equal itself.

If it is non-zero, and equals itself, then it is posisble to find a
smaller number less than the infinitesimal. How was this issue
resolved in surreals?

Secondly, if it equals itself, then it doesn't mirror the essential
property of the infinity.

In addition, surreals also
allow for numbers that are closer to a real than any other real, ex.
124 + (1/omega) is closer to 124 than evey real number. What do you
require for it to be a "symmetric counterpart" and why is this a good
thing?

We need not create any additional symmetry that what already exists in
nature. We just need to preserve it. What I saw is - the "point and
set" modeling of the continuum is destroying the symmetry, and making
jumps in the natural processes. For example, while dividing a line
segment with finite number of cuts, what we get is regions separated
by boundaries. The regions and boundaries are mutually opposite
structures and can only exist together.

Now, continuing the cutting process endlessly, the "set and point"
model suddenly eliminates the regions from the scene and calls
everything as boundaries or points. The justification given was that
the regions got to small to distinguish from boundaries. But my
contention is that regions only got "too small" but not zero extents,
while boundaries always are of zero extent. This big difference is
ignored in the model and it destroyed the symmetry leading all logical
paradoxes and theoretical patch work to cover it. The asymmetry is
very glaring can be demonstrated by using common sense, as I have
detailed in another thread.


Anyway, surreal numbers are rigorously defined, your "b" (do you have
a better name? Does "b" stand for something?)

No, "b" doesn't stand for anything. It just happens to be the second
alphabet, so maps to "second" number after zero.

- venkat
.

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