Hyperreal Properties: Understanding the Archimedean Property & Superstructures

  • Thread starter homology
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In summary, internal subsets of the hyperreals can be described by the power-set operation, but the set-theoretic power-set is not always able to give you an external subset.
  • #1
homology
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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
 
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  • #2
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.
 
  • #3
I'm very curious about this "Numbers"?..in fact could you treat expression using hyperreal numbers in the form:

[tex] \infty^{3}+\infty^{2}-\infty [/tex] or define

[tex] exp(\infty^{3})-sin(\infty)+|\infty| [/tex]
 
  • #4
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, [itex]+\infty[/itex] serves the exact same role in nonstandard analysis as it does in standard analysis, and thus expressions like

[tex] \infty^{3}+\infty^{2}-\infty [/tex]

are undefined.


Externally speaking, there are infinite hyperreals ([itex]+\infty[/itex] is not a hyperreal), and you can do all the arithmetic you want with those.
 
  • #5
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
 
  • #6
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.
 
Last edited:

1. What is the Archimedean property?

The Archimedean property is a mathematical concept that states that for any two real numbers, there is always a third real number that lies between them. This property is essential in defining the concept of distance and continuity in mathematics.

2. How does the Archimedean property relate to hyperreal properties?

The Archimedean property is a fundamental concept in understanding hyperreal properties. It helps to define the distance between infinitely close numbers, which is essential in understanding the concept of infinitesimal numbers in hyperreal analysis.

3. What are superstructures in hyperreal properties?

In hyperreal analysis, superstructures refer to sets of numbers that contain both real and infinitesimal numbers. They are used to extend the real number system and allow for the representation of infinitely small and infinitely large quantities.

4. What are some applications of hyperreal properties?

Hyperreal properties have various applications in fields such as physics, economics, and computer science. They are used to model complex systems and problems that involve infinitesimal and infinite quantities, such as in calculus and differential equations.

5. How do hyperreal properties differ from traditional real numbers?

Hyperreal properties differ from traditional real numbers in that they include infinitesimal and infinite quantities. This allows for a more flexible and precise approach to mathematical analysis, particularly in dealing with problems involving infinitely small or large quantities.

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