Simple theorem in the hyper-reals (reals with infinites and infinitesimals)

[Lance_Dirk]

Note on notation: This seems pretty standard, but just to be clear, for any function F defined on the reals (R), *F is the extension of F to the hyper-reals (*R). This basically means that any logical sentence true of F in the reals is true of *F in the hyper-reals. If you're familiar with the hyper-reals, you should be aware of this relation.

I need to prove the following:
Let F: A -> R where A is some subset of R, and let F be one-to-one. We've already shown this implies *F: *A -> *R. Show that if we have an x such that x is in *A but x is not in A, then *F(x) is not in R (but of course it is in *R).

This makes intuitive sense to me but I have no idea how to show it.

Any help is appreciated, even if you can't totally solve it. Just showing me another avenue of thought will surely be helpful.

Thanks in advance.

 

[daon2]

Do you know anything else that is helpful? I am not too familiar with this set, I just know what it is.

Is f onto or continuous? Does f being 1-1 or continuous imply f* is? If f* a unique extension?

 

[Lance_Dirk]

Do you know anything else that is helpful? I am not too familiar with this set, I just know what it is.

Is f onto or continuous? Does f being 1-1 or continuous imply f* is? If f* a unique extension?

F is not necessarily onto or continuous. If it was either of those I think I would now what to do. I'm pretty sure that F being one to one implies that *F is. And I know that *F is unique.

One thing I can say is that A must be infinite, since the subsequent problem asks to show that A = *A iff A is finite. I also am almost totally certain that any x in *A which is not in A will be either infinite, infinitesimal, or a real plus or minus an infinitesimal. In other words, x in *A and not in A implies x in *R and x not in R.

 

[daon2]

Then I think the solution will be based on how the function is extended, which I have little knowledge of. Though I did just get a hyperreal baby-sized education from wikipedia.

It seems real numbers are those sequences of the form (?,a,a,a,...). That is, "eventually constant." And the "natural extention" of a function with domain A to *A sends a sequence in A: (a,b,c,...) -> (f(a), f(b),f(c),...) which is certainly going to be a real number if and only if the sequence (f(a),f(b),f(c),...) is eventually constant.

So if x belongs to *A but not A, then x is a sequence which is not eventually constant. Since F is 1-1, this implies F(x) is also a sequence which will never be constant. There are details needed to be shown, but I think that is the proof.

 

[Lance_Dirk]

Then I think the solution will be based on how the function is extended, which I have little knowledge of. Though I did just get a hyperreal baby-sized education from wikipedia.

It seems real numbers are those sequences of the form (?,a,a,a,...). That is, "eventually constant." And the "natural extention" of a function with domain A to *A sends a sequence in A: (a,b,c,...) -> (f(a), f(b),f(c),...) which is certainly going to be a real number if and only if the sequence (f(a),f(b),f(c),...) is eventually constant.

So if x belongs to *A but not A, then x is a sequence which is not eventually constant. Since F is 1-1, this implies F(x) is also a sequence which will never be constant. There are details needed to be shown, but I think that is the proof.

Yeah, upon looking into the hyper-reals, I found a definition of them which involved expressing each hyper-real as a sequence (well, presumably the limit of the sequence, right? I need to look closer.) Given this definition, I see how this theorem would make more sense. However, the book I've been assigned to use has no such expression, so I want to try to figure out a way to do it that doesn't rely on this expression. In fact, my book hardly discusses extensions of functions at all, so I'm thinking there must be a simpler way of looking at them that I'm missing. Maybe I can show that each hyper real is the limit of some sequence of reals? I don't know. But as I said, any alternative view of the problem is helpful, so you have my profuse gratitude, whatever that means.

 

[daon2]

Yeah, upon looking into the hyper-reals, I found a definition of them which involved expressing each hyper-real as a sequence (well, presumably the limit of the sequence, right? I need to look closer.) Given this definition, I see how this theorem would make more sense. However, the book I've been assigned to use has no such expression, so I want to try to figure out a way to do it that doesn't rely on this expression. In fact, my book hardly discusses extensions of functions at all, so I'm thinking there must be a simpler way of looking at them that I'm missing. Maybe I can show that each hyper real is the limit of some sequence of reals? I don't know. But as I said, any alternative view of the problem is helpful, so you have my profuse gratitude, whatever that means.

They are not limits of sequences (limits are either defined or not defined), but rather the sequences themselves, at least according to the construction on wikipedia. For example 5 -> (5,5,5,...) is a hyper-real number but is, by this embedding, a real number (who's limit is 5). The infinitesimals are those which have a limit of 0 (though apparently there could be infinitesimals which do not have a limit of 0).

Example: x=(1,1/2,1/3,1/4,...) is an infinitesimal, and 1/x is an infinite, both contained in this field.

Second, y=(6,4,3,1,1/2,1/3,1/4,...) is a hyperreal infinitesimal that is equal to x, as they are eventually "the same"

By this construction, your proposition is easily proved. Being that this is the only one I am aware of, I don't know how else it could be done.