Re: The infintely small number b

On Nov 19, 11:42 am, Joe Blow <justarandomi...@xxxxxxxxx> wrote:
The object with the properties you describe, whatever it is, would necessitate that the space in which it is contained be non-Hausdorff. Much of mathematics depends explicitly on the fact that the real line has the Hausdorff property, and when you throw it out you're left with an object that isn't terribly useful.

So, while you might be able to define a space which has some of the properties you want, the question is why would you want to? What would you be able to study on this awkward space that you can't adapt the well-studied space of real numbers to do?


The introduction of b and its strange operations might have led you to
believe that I'm trying to introduce some non-standard numbers.
However, b is not the core part of my thoughts. I think I have seen a
core problem which is manifesting itself as different issues in
different areas.

The core issue: Assumption that a line segment is made of distinct
points.

Its various forms:

- Zero, which stands for non-existence of something, can build up a
finite extent, when added to itself infinite times.

- A line segment or an interval of a continuum can be modeled using a
set of distinct real numbers.

- In the above scenario, real numbers are seen as points instead of
extents.

- Boundaries can exist as standalone objects independent of regions.

- The intersection point of two lines is *part* of those lines.

- A point can lie in one of the three areas - inside a region, on the
boundary, outside the region.

- The greatest and smallest numbers may or may not exist in a set of
ordered real numbers representing a finite interval, depending on
whether we chose to include its boundary or not, though it doesn't
make any difference to the interval itself. When you say that boundary
is not a member of an open interval, then what is largest number which
now serves as the boundary of this set? If you don't even know it's
inclusive boundary, have you actually modeled the interval?

- We can accept an infinitely large positive number but can't accept
infinitely small positive number.

- A single real number can have zero or more decimal representations.
We knew every real number has dual decimal representations, but what
is the largest x<1? I think it doesn't have any decimal
representation.

- several more such forms of the same issue.

The number b came up as a symmetric counterpart of infinity with
similar properties, that can serve as smallest positive extent,
instead of collapsing it to zero. I do not intend to introduce any new
number system here. Any number system which accepts infinity as a
valid member can easily accept b as a member as well.

venkat

.

Relevant Pages

  • Re: The infintely small number b
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    (sci.math)
  • Re: The infintely small number b
    ... b is not equal to b in arithmetic sense (addition, subtraction). ... The infinitesimal is a generalization of quantity. ... You cannot subtract infinity. ... their own finite set of boundaries. ...
    (sci.math)
  • Re: Einstein Made It Up As He Went Along
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