'Unlimited' hyperreal numbers

**Suni** · Jan14-06, 08:40 AM

Hi all
I've seen the term 'unlimited' and 'infinity' used interchangeably in non-standard analysis. However when reading a scientific american magazine (Nov 1994 - "Resolving Zeno's Paradoxes") the author says this:

Because an infinitesimal is very small, its inverse will be very large (in the standard realm, the inverse of one millionth is one million). This type of nonstandard number is called an unlimited number. The unlimited numbers, though large, are finite and hence smaller than the truly infinite numbers created in mathematics. These unlimited numbers live in a kind of twilight zone between the familiar standard numbers, which are finite, and the in-finite ones.

I am slightly confused. Which is it? I can't see why the inverse of an infinitesimal would be infinite if 'unlimited' and 'infinite' actually had the same meaning.

**matt grime** · Jan14-06, 09:17 AM

All of the words seem scrambled. I have no idea what the author even means by 'truly infinite numbers' that are 'created in mathematics'.

He can't possibly mean infinite cardinals can he? In any case, since the reicprocals of hyperreals are not cardinals the comparison doesn't make sense.

(Conway's surreals may be different)

**cogito²** · Jan14-06, 09:28 AM

The writers use the axioms and results of Internal Set Theory (developed by Edward Nelson; link) so you have to take that into account. The results of the theory are only true when using the same axiom system. Since you've probably never worked with IST (I never have; in fact, I hadn't even heard of it until I read that article after seeing your post) the concepts probably seem as odd to you as they do to me, but it all works out logically (I presume) inside IST. So basically, your questions cannot really be answered without actually using IST. If you want to understand it a little better, the wikipedia link I gave is probably a good place to start.

**cogito²** · Jan14-06, 09:44 AM

Here is Edward Nelson's homepage and here is a chapter on Internal Set Theory (linked on his homepage) that he has not finished.

**Suni** · Jan14-06, 09:50 AM

Thanks for the replies guys!

Yes cogito² i think you're right.. ive mistaken this IST business with the hyperreal set and its implications. It actually looks like the whole process is undertaken in the Real set itself. Looks like i have a lot of reading to do.

The first half of the wikipedia article actually appears to be borderline philosophy.

**Hurkyl** · Jan14-06, 12:38 PM

Even in the ordinary framework, there is a sense in which the author is correct... every hyperreal is internally finite.

However, this is confusing, because "finite" has a specific meaning in our external analysis. Therefore, we refer to "internally finite" as "hyperfinite".

The notion of hyperfiniteness arises from the hypernaturals: something is hyperfinite if there exists a hypernatural with greater magnitude. (Remove "hyper" from everything and you have the usual meaning of "finite")

Maybe this picture can help explain the "twilight zone":

When doing real analysis, we like to consider the extended reals. (And the extended naturals). The set of extended reals are defined by $LaTeX Code: \\bar{\\mathbb{R}} := \\mathbb{R} \\cup \\{-\\infty, +\\infty\\}$ -- in other words, we simply add two new symbols to the reals. These symbols are incorporated into the ordering <, by saying that $LaTeX Code: -\\infty$ is the smallest, and $LaTeX Code: +\\infty$ is the biggest.

While in your elementary calc classes, you were taught that when these symbols appeared in formulas, it was just a symbol and didn't really mean anything -- well, if you so choose, you can generally reinterpret all of those formulas so that it works with these two infinities that live in the extended reals. (So, for example, saying the limit of something is $LaTeX Code: +\\infty$ means essentially the same thing as saying the limit is

)

Allow me to draw a picture of the extended naturals in order:

$LaTeX Code: 0, 1, 2, \\ldots | +\\infty$

Where I've used the pipe (|) for "grouping" -- everything to the left forms one group (the naturals).

Well, all of the unlimited hypernaturals are inserted "before infinity". Here is a picture of the hypernaturals, in order:

$LaTeX Code: 0, 1, 2, \\ldots | \\mbox{ unlimited hypernaturals } | +\\infty$