Re: how to list all of the real numbers

On Sep 5, 1:11 pm, Jonathan Hoyle <jonho...@xxxxxxx> wrote:
On Sep 5, 10:15 am, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:



Lies!

Generally in mathematical discussion the infinitesimal means the
differential.

Nothing can be further from the truth. No mathematician uses the word
in that way, not even those doing Non-Standard Analysis. Ross, it
appears you are not even aware of what mathematicians do.

When was the last time you spoke with a mathematician? Any local
university should have your pick of them, and you will be quickly
disabused of this idea in a single discussion.

That's where, for example, in physics: mathematics of a
physical system will often be considered in terms of infinitesimals,
each differential, in the system.

I cannot speak for physicists, but it was my understanding that they
used Calculus properly as well, not in your 300 year old anachronistic
way. If there are those doing so in that fashion, then they do so
without a mathematical basis.

About the geometry, and the geometrical interpretations of the real
numbers, consider how for example a real number can be a line
segment on the axis or any point on a line, geometrically, where lines
are defined in terms of points. Fluxion infinitesimal means
mathematical objects with the fluxion properties of the infinitesimal.

Again, I would be rather surprised to hear of any modern physicist
even knowing what the word "fluxion" meant, let alone using it in its
original (and moot) sense.

I'm happy to tell you that I have logical replies, phrased in terms of
standard logic with their standard meaning, to help illustrate why I
am not incorrect, in the mathematical sense. So, when I say that the
hyperreals are the reals, I as well have a formal mathematical
argument to that effect.

No, you do not. The reals are Dedekind complete, whereas the
hyperreals are not. Thus, they cannot be the same. This is not
opinion, this is fact.

Hope that helps,

Jonathan Hoylehttp://www.jonhoyle.com

The reasoning towards that the hyperreals are a subset of the reals is
that in the notion of hyperreals basically indicating a halo of
elements about a real number, given the completeness of the reals
(gaplessness, LUB property), that those hyperreals would only have
geometrical meaning as points on a line, which are real numbers, and
that they do have geometrical meaning, else they wouldn't.

That's where the real numbers are totally ordered, and for the
hyperreals to be among them, they are so as well, because of the
gaplessness of the continuum of real numbers. With the notion that
non-real hyperreals map the neighboring points of a real number, that
real number already has all its neighbors and they're all real
numbers.

(So, I advocate a post-Euclidean geometry.)

Does the limit of the converging infinite series equal the sum of the
partial terms' differences? For no finite nonconstant series does
it. It is only in the infinite that it does.

Generally in discussion the infinitesimal refers to the differential.
Also, generally differential means arbitrarily small, and non-zero,
and the sum of dx for each real from zero to one equals one.

Ross

--
Finlayson Consulting


.

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