Re: An online source for a hyperreal primer?

(Only posted to sci.math, as this part of the thread is off-topic for
any of the other newsgroups.)

On Oct 8, 3:15 pm, lwal...@xxxxxxxxx wrote:
On Oct 7, 12:58 am, Proginoskes <CCHeck...@xxxxxxxxx> wrote:

On Oct 6, 9:32 pm, lwal...@xxxxxxxxx wrote:

[...]
Although I know that AP has already rejected my use of the
hyperreals to describe his p-adics, I can still formalize what
he's saying in the hyperreals. [...]

Is there a good primer on the hyperreals online? What you're posting
looks interesting, but I have the feeling I'm missing out on some of
the details.

Here is the Wikipedia page:

http://en.wikipedia.org/wiki/Hyperreal_number

I glanced at this page but couldn't grok it right off. Then I moved to
MathWorld's hyperreals page but was *very* disappointed.

and the Wikipedia page has two links near the bottom:

http://www.math.wisc.edu/~keisler/chapter_1a.pdf (1.6 MB)
http://mathforum.org/dr.math/faq/analysis_hyperreals.html

(Actually, there are three links, but the third is hardly
for beginners at all -- it involves the existence of a
countably saturated extension of the reals.)

Notice that these links focus more on the infinitesimals
and their usefulness in calculus, but AP's p-adics are
more like infinite numbers than infinitesimals.

Don't tell him that. 8-) His argument that "there is only one
infinity" is based on the assumption that 1/infinity is zero, when it
should actually be an infinitesimal.

I read the "Chapter 1a" link, and have a few comments.

(1) The Transfer Principle looks like voodoo mathematics to me. In
fact, it seems to lead to a contradiction (based on the idea whether
hyper-hyperreals exist). I'm sure it's not, and I can even see where
I'm going wrong, but not why.

Here goes: Consider the statement

(Ee)(Ax in R)(0 < x ==> e < x)

which is a basic axiom of hyperreals; namely, that there is at least
an infinitesimal other than 0. Now let's use the Transfer Principle on
it. (I'm sure this is where the error is being made, but I don't know
why, based on what I've read.)

(Ed)(Ax in R*)(0 < x ==> d < x)

which guarantees that this statement is true. So let d be as given (a
positive number of some type that is less than any positive
infinitesimal). Now d is less than any positive real number, so d is
an infinitesimal. Hence the statement above is true when
x = d, which means d < d. Contradiction.

(2) The "Standard Part Principle" states that "Every finite hyperreal
number is infinitely close to exactly one real number." Isn't this
provable? Certainly, there is at most one real number which can
satisfy the above condition, as no real number is infinitely close to
any other real number.

To show that there is a real number that satisfies the condition, let
S be the set of all real numbers less than a given hyperreal b; then
take the supremum of S (which is a real number, since the set S is
bounded above by any upper bound of b).

(3) Is there a way to generate ALL of the infinitesimals? The paper I
read will define new infinitesimals in terms of the one that is given
to exist, but I can't get a grasp on when you know that you have them
all. (From what you've posted, along with a brief look at the Dr. Math
page, it seems that you can generate infinitesimals from sequences of
real numbers, which suggests that there might be more of them than
real numbers.)

(4) The definition of the derivative which sticks to the real numbers
looks intuitive to me, but the definition which uses hyperreals looks
like it needs some justification. Why should you choose the standard
part of the difference quotient? (I'm looking for an answer other than
"because it works in all the simple cases.")

(5) I was reminded of the following quote: "In this book it is spoken
of the sephiroth and the paths, of spirits and conjurations, of gods,
spheres, planes and many other things which may or may not exist. It
is immaterial whether they exist or not. By doing certain things
certain results follow; students are most earnestly warned against
attributing objective reality or philosophical validity to any of
them."

--- Christopher Heckman

.

Relevant Pages

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  • Re: What is the angle of a circular arc?
    ... to reformulate the Leibniz's ideas of infinitesimals ... reals or hyperreals depends on whatever is easier. ... Look up 'Non-standard analysis' ...
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