Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/18/05
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Date: 18 Jan 2005 02:22:10 -0800
If you select positive integers a,b to form the rational a/b, then half
of that, a/2b, is also a rational positive number that is less than
a/b. Neither of them is iota.
If you consider that infinitesimals, which some equate to zero, are
positive, then as the real numbers are continuous, then the
infinitesimals in the neighborhood of zero are real numbers, because
the real numbers are continuous, and not just an Abelian group, ring,
and field, with a multiplicative identity. If there exists positive
and non-zero infinitesimals in the field of the real numbers and it is
a field then the real numbers also include a "point at infinity", or
otherwise the multiplicative inverse of each element of the reals is
contained in the reals.
What's 1/0?
If the reals include iota and are a field then iota's multiplicative
inverse is oo, or I, iI = 1, 2iota's multiplicative inverse oo/2,
etcetera. Where there are problems defining iota there will probably
be issues defining half infinity.
Where that is so then there is the consideration of iota/2. One notion
is that the division of iota is undefined, or iota/x = 0 for x > 1.
The idea and definition of sorts of iota is that it is the least
positive real number.
You ask what iota/2 is, my considered response is that it is plainly
not to be divided by two, if the result of iota/2 were to be itself be
positive then the original term was mislabelled and was not actually
iota.
That leads into the obvious notion that the infinitesimals do not exist
in the real numbers and that the integral multiples of iota are a
constant zero.
While that is so the reals are continuous, and indeed there are points
between zero and any 1/n for finite integer n, and that returns to the
notion that there are positive, non-zero infinitesimals. The real
number line is continuous. It's nice to know that there exist real
numbers larger than zero.
One thing to consider is that iota, an infinitesimal, is an element of
the reals, but that it is not in the same field or ring structure as
the elements defined as solutions to polynomials with finite integer
coefficients that are not complex or continued fractions of finite
integers, or transcendentals defined as a converging value of an
algebraic, that the set of reals is comprised of two or more field
structures, perhaps indeed one for each element of the normal field,
and that that would apply to other similarly typed structures.
If somebody argues that there are no infinitesimals in the reals, then
I say there are infinitesimals in non-standard models of the reals, and
then say that they are the real numbers.
What do you think about that, the implicit fields with different
operations? I've been complaining about Dedekind and Cauchy's
insufficiency. If you talk about "non-computable" reals, so have you.
If the normal ordering of the non-negative reals is a well-ordering,
then that's a particularly useful way!
Regards,
Ross Finlayson
-- "Skolemize: your model is countable."
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