Hyperreal Properties: Understanding the Archimedean Property & Superstructures

[homology]

Hey folks,

I've been playing around with these (hyperreals) lately and have a couple questions. One concerns the Archimedean property. I keep finding different formulations of it. Some of them lend themselves to transfer. For example, if the AP is that for every real x there exists positive n such that |x|<n then this transfers nicely. Another formulation stipulates that n is finite which wouldn't transfer. Anywho, I'd like to get the sharp, orthodox take on this.

Question numero 2: So I've seen references to using superstructures and constructing a first order logic on the superstructure so you can talk about things like measures and what not. Now the completeness property of the reals (every bounded subset has...) seems as though it could be phrased in such a logic and then transferred?

I'm sure these questions reveal my naive understanding of the hyperreals. Math gods, be gentle...

thanks

 

[Hurkyl]

Another formulation stipulates that n is finite which wouldn't transfer

"n is finite" does transfer. In (what I think is) the usual language, it transfers to "n is hyperfinite".

Of course, the definition of "finite" is usually "smaller in magnitude than some natural number".

Now the completeness property of the reals (every bounded subset has...) seems as though it could be phrased in such a logic and then transferred?

Yep. Every bounded internal set of hyperreal numbers has a least upper bound, et cetera.

 

[lokofer]

I'm very curious about this "Numbers"?..in fact could you treat expression using hyperreal numbers in the form:

$$\infty^{3}+\infty^{2}-\infty$$ or define

$$exp(\infty^{3})-sin(\infty)+|\infty|$$

 

[Hurkyl]

Essentially everything you can do with the real numbers, you can do with the hyperreal numbers. Everything you can't do with the real numbers, you can't do with the hyperreal numbers. (Though in both cases you have to be careful about doing things "internally" and not "externally")

In particular, $+\infty$ serves the exact same role in nonstandard analysis as it does in standard analysis, and thus expressions like

$$\infty^{3}+\infty^{2}-\infty$$

are undefined.


Externally speaking, there are infinite hyperreals ($+\infty$ is not a hyperreal), and you can do all the arithmetic you want with those.

 

[homology]

Thanks Hurkyl

Thanks Dude,

Another question oh fountain of mathy knowledge. Could you give me a quick breakdown of internal and external? Just rough is fine. I see that I've missed something important.

cheers,


Kevin

 

[Hurkyl]

The dictionary definition is that something is internal if and only if:

(1) It is the *-transfer of something standard
(2) It is an element of an internal set

(so this is a recursive definition)




I prefer a more "constructive" interpretation, though -- something is internal if and only if you can describe it analytically. If you must resort to something else (such as comparing the reals to the hyperreals, or invoking pure some pure set-theory), it's external.

In particular, the analytical power-set operation is not the set-theoretic power-set operation. (I think that was my biggest stumbling block in trying to understand this stuff) The analytic power-set operation only gives you internal subsets -- but the set-theroetic power-set is capable of giving you any external subset.

The standard model of analysis is the only one in which the analytic power-set coincides with the set-theoretic power-set. In any nonstandard model, the analytic power-set is "missing" some sets.

For example, the set of all infinitessimals is not an element of the (analytic) power-set of the hyperreals.


P.S. if you're algebraically minded, you might try studying the (related, but simpler) theory of real closed fields first; for me, that really illuminated the ideas from formal logic that are at work here.