Re: a theory of countable reals
- From: "toshiaki" <farawfu@xxxxxxxxx>
- Date: Wed, 25 Jul 2007 10:54:46 +0900
There is an argument that pi does not exist. Pi cannot be
represented by geometrically. But this is inevitable in
mathematics. We can represent many values using pi like the
probability density function, and calculate many values
related to a circle.
These calculation cannot be done by its approximation.
After equation is established, we can calculate its value
in any detail.
Even if pi cannot be constructed on a line geometrically
or with decimals, we can deal with it as if it is there.
Given a circle, we can think about the ratio of the
circumference to the diamiter.
If it can be approximated in any detail, we can assume
its existence as black box regardless of the
representation and can use in various calculation. This
black box can be approximatid unlimitedly to the ratio
different from using one of the approximations.
I think that the infinite set like N could be used in
this way witout assuming a size. All infinite objects
we can use is countable. "countable" is not a size.
"countable" is constructibe or expressible or
specifiable.
It is not certain that a line is made of points. If it
is made of distinguishable points, this cannot be
realized without gap corresponding to the difference
between them.
When it comes to the calculation of a circle, the problem
becomes clearer. We cannot calculate the circumference of
a circle without subdividing it and taking the limit.
An existence of a circle made of smooth and connected
line is based on the assumption.
.
- References:
- a theory of countable reals
- From: toshiaki
- a theory of countable reals
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