Hyperreals and Infinitismal Calculus

nonnus29

A Hyperreal, $<math>\mathbb{^*R}</math>$ is a extension of the reals $<math>\mathbb{R}</math>$ such that if $<math>x \in \mathbb{^*R}</math>$ then:

$<math>|x| < a</math>$ for all $<math>a \in \mathbb{R}</math>$ is an infinitismal Hyperreal
$<math>|x| > a</math>$ for all $<math>a \in \mathbb{R}</math>$ is an infinite Hyperreal
$<math>a < x < b</math>$ for $<math>a,b \in \mathbb{R}</math>$ is a finite Hyperreal.

This is saying that a Hyperreal can be less than any Real number or greater than any Real number.

Hyperreals are used in "nonstandard" Analysis to give a basis for Calculus in place of limits:

$<math>|\epsilon - \delta| = L</math>$

Fluxions/Infinitismals are how Newton and Leibniz justified that Calculus worked, but they didn't have a formal definition of what it actually was. That came in 1960 with the definition of the Hyperreals.

What do you guys think of Hyperreals? I think it seems like a bunch of hooey. If there were Hyperreals, that would be like saying there IS A LARGEST INTEGER n! And there is no infinite!

CGamesPlay

I'd like to see the proof for the existence of a finite hyperreal, the others don't seem innovative or useful, though.

[append]

Well, a < x < b, a < b for a, b in R: x is a finite Hyperreal.

StevenVI

It sounds similar to denormalized numbers in IEEE 754.

I'm sure that your definitions meant to have absolute values on them. I hope you first have gone through the proofs using limits before exploring into weird territory.

In response to your query, this definition, to me, sounds like it is redefining what a real number is. At the very least, they are redefining the concept of a "number." A real number, when expanded, has infinite decimal places. It might be an infinite number of zeros, but they are still there. If the hyperreal number can also be represented with digits (regardless of base), then they must be real numbers as well. Of course, this contradicts the definition for hyperreal numbers.

(In other words, I am unsure how to interpret the comparisons without more definitions. I am forced to assume that the comparisons are of numbers with no imaginary component. This leads me to my contradiction.)

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This reminds me of an interesting thing I was once told: Math requires faith, just as any other religion does. Faith that the constructs are consistent. I will demonstrate this with some set theory.

Say that a set is "weird" if it contains itself. For example, the set $<math>A=\{A, X, Z\}</math>$ is weird because it contains itself.

Let S be the set containing all non-weird sets. Suppose that S is not weird. Then $<math>S\in S</math>$ . However, that implies that S is weird. Thus it must be concluded that S is weird, so $<math>S\not\in S</math>$ . Again, though, this implies that S is NOT weird. The definition is inconsistent.

Edit: ehh, I might as well point out the inconsistency. The definition implies that all sets are either weird or not weird. The way to patch this up is to add room for sets which are neither weird nor non-weird (or both weird and non-weird), similar to the way a set can be neither open nor closed (or both open and closed).

I believe it was Gödel who showed that you cannot prove a system to be consistent.

Goalie Ca

Most mathematicians would agree we are build everything from Zermelo–Fraenkel set theory.

There are numbers bigger than infinite.
http://en.wikipedia.org/wiki/Aleph_number

The infinitesimals are better described using epsilon-delta definition of a limit. Likewise, for abstract algebra and vector spaces there are similar constructs. I call them pertubations.

The axiom of choice is what i have a real problem with.
http://en.wikipedia.org/wiki/Axiom_of_choice

nonnus29

Harry Carey, I think what you described is the Russell's paradox. And yes, I believe I did leave out the abs value symbols on mistake.

Quote:

The infinitesimals are better described using epsilon-delta definition of a limit.

The point of Hyperreals is to give Calculus a foundation WITHOUT epsilon-delta arguments. To provide a logical foundation to Newton and Liebniz intuition I suppose. See Chapter 1 of the book below:

.http://www.math.wisc.edu/~keisler/calc.html

Might be of interest to H.C. as well...

Karadoc ~~

Goalie Ca said:

There are numbers bigger than infinite.
http://en.wikipedia.org/wiki/Aleph_number

There are only "numbers bigger than infinite" if you use the language is a loose kind of way. In a stricter sense, those Aleph thingies are not talking about the same kind of stuff as "infinity". That wikipedia article says this:

Quote:

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

Thomas Harte

Quote:

What do you guys think of Hyperreals? I think it seems like a bunch of hooey. If there were Hyperreals, that would be like saying there IS A LARGEST INTEGER n! And there is no infinite!

They sound like a useful construct and seem to be perfectly well defined, so I have no problem with them. Obviously I couldn't have a hyperreal number of oranges or anything like that, but I can't have an imaginary number of them either.

William Labbett

If hyperreals can tell you the volume of a sphere I think they're great.

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