Complex Numbers in Non-Standard Analysis

[JungleJesus]

In Non-Standard analysis, the "real" numbers are extended by adding infinitesimal elements and their reciprocals, infinite elements. These numbers are referred to as hyperreals and are logically sound and analytically rigorous. When one considers the "Standard Part" function st(x), one can define derivatives, integrals, limits, and all other aspects of real analysis in a logically concise and intuitive way.

That being said, have the hyperreal numbers been extended to the complex number system? If so, where can I find a complete, exhaustive resource? I've been looking for a long time and am disappointed to find the term "hypercomplex number" has already been taken (single tear).

Please forgive me if I'm repeating someone else's question. I couldn't match any of the other posts.

 

[g_edgar]

That being said, have the hyperreal numbers been extended to the complex number system?
yes
where can I find a complete, exhaustive resource?
This does not follow from the first answer. I think there is one chapter on this in Robinson's book.

 

[JungleJesus]

Has anyone taken this into serious consideration, or is Non-Standard Complex Analysis a relatively untouched field of study?

 

[Bacle]

JJ: I hope I am not hijacking your thread.

I am just curious on knowing of some result that holds for non-standard

real numbers but does not hold for standard reals. One difference I know

is that the non-standard reals are non-Archimedean (just a formal way of

saying they admit infinitesimals ), while the reals are not. I also know that

the non-standards are elementarily-equivalent to the standards, so we would

need to find a second-order-language statement in the non-standards.

Anyone know of any such example?

 

[JungleJesus]

I am just curious on knowing of some result that holds for non-standard

real numbers but does not hold for standard reals. One difference I know

is that the non-standard reals are non-Archimedean (just a formal way of

saying they admit infinitesimals ), while the reals are not. I also know that

the non-standards are elementarily-equivalent to the standards, so we would

need to find a second-order-language statement in the non-standards.

Anyone know of any such example?

There are many statements which hold for reals and not for hyperreals and vice versa. The general rules of thumb are the Extension Principle and the Transfer Principle. These state (in its simplest form) that any "real statement" involving real expressions will hold when applied to the hyperreal numbers. A real statement is any combination of equations or inequalities about real expressions, as well as any statement specifying whether or not a real expression is defined.

For example, the closure law for addition and the order axioms hold for the hyperreals. However, statements which are not real expressions need not "transfer" to the hyperreals. A good example is the definition of an interval for the real numbers. Certain definitions of intervals on the real number line do not apply to the hyperreals.

It is also impossible for two real numbers to be "infinitely close" to one another. On the contrary, for any given hyperreal number, there are infinitely many other hyperreals which are infinitely close to it.

I'm not completely familiar with the logical formalism behind the hyperreals. As far as second-order logic is concerned, I would start with some model theory, which Robinson developed/used while constructing the hyperreals.

 

[Hurkyl]

The basic idea of constructing the hyperreals from the reals can be extended to the entirety of "real analysis".

So, there are non-standard versions of things like manifolds, measures, most topological spaces one would encounter in "real life", most algebraic structures one would encounter in "real life", and so on.

And, of course, the transfer sill applies. Any statement in the language of analysis is a theorem of the standard model iff it is a theorem of the non-standard model.


In the set-theoretic aspect, the main difference (and IMHO the hardest to wrap your head around) is as follows:

• Let S be a set in the non-standard model
• Let P be the standard power set operator
• Let *P be the non-standard power set operator

If S is infinite by the standard measure of cardinality, then

$${}^\star \mathcal{P}(S) \subset \mathcal{P}(S)$$​

in particular the inclusion is proper; there are standard subsets of S that are not sets in the non-standard model.


This is how you reconcile things like the statement "The real numbers are the only complete ordered field". The transfer principle applies, so the hyperreals are a complete ordered field in the non-standard model. But we know from set theory that the real numbers are the only complete ordered field! What gives?

Translating the transfer of, for example, the least upper bound property into standard language, the relevant theorem is:

If S is a nonempty bounded subset of the hyperreals that is a set of the non-standard model, then S has a least upper bound​

We use the term "internal" and "external" to distinguish between the two ideas. The hyperreals are internally complete -- they satisfy the internal version of completeness stated above. However, the hyperreals are externally incomplete. For example, the set of infinitessimals does not have a least upper bound.

(This counts as a proof that the set of infinitessimals is an external set and not an internal set -- it is not part of the non-standard model)


I'm rambling but I hope it gives useful knowledge to both of your questions -- the presence of manifolds and differential geometry for the OP, and the difference between internal and external for the hijacker.

 

[JungleJesus]

Ahh. So the Transfer principle allows the complex structures to be defined for the non-standard model. Very interesting.

Thank you, Hurkyl.

Does anyone know where I could find an online book or something along those lines? I know Robinson briefly mentioned the complex extension in his book, but that's about as much as I have seen.

 

[Bacle]

I am bit confused; I wonder if I misunderstood --or misunderestimated--what you said,

Hurkyl; I am also unclear about something JJ said:

I know the term Transfer Principle, by the name of "Elementary Equivalence", which

states that for elementary-equivalent models (or is it structures?)M1, M2 ; where

M2 contains M1 , every 1st order statement that is true in M1 is also true in M2.

Now, in our case, the standard reals R are M1, and the hyperreals are M2, so every

1st-order statement that holds in M1, would hold in hyperreals. If I understood you

well, you are going in the opposite direction.

Also: aren't the standard reals the only _Archimedean_ ordered field ( and not just the

only ordered field). Do you mean order-complete (which I guess would mean they

satisfy the lub property)?

I thought that the Archimedean property of Standard R was one of the key aspects that

does not translate into hyperreals.

JJ : I don't see how you can use the transfer principle, or elementary equivalence

to define non-standard complexes. Would you explain what you mean, please?

not translate into the

 

[JungleJesus]

A standard complex number can be conceptualized as a pair of real numbers (x,y) of the form "x + iy" where i is the square-root of -1. I am wondering if/how the non-standard complex numbers can be constructed.

By my intuition alone, I would construct the non-standard complex number with a pair of hyperreal numbers (X,Y) of the form "X + jY" where j is the non-standard square-root of -1 (the non-standard square root function f* is just the natural extension of the regular square-root function). j is basically the same thing as i, but I made the distinction just in case.

Once the non-standard complex number exists, one can implement higher-level structures like functions, spaces, and analysis.

 

[Hurkyl]

Now, in our case, the standard reals R are M1, and the hyperreals are M2, so every

1st-order statement that holds in M1, would hold in hyperreals. If I understood you

well, you are going in the opposite direction.

That is right -- elementary equivalence is an "if-and-only-if" statement. This follows from the one-directional version because if any particular statement S holds in M2, then "not S" cannot hold in M1.


Now, "reals" and "hyperreals" are, in some sense, the simple version. The grander method of non-standard analysis is to construct a formal theory that let's you describe not only algebra, but enough of set-theory to describe analysis too.

Also: aren't the standard reals the only _Archimedean_ ordered field ( and not just the

only ordered field).

Yep -- and the hyperreals do, in fact, satisfy the transfer of the Archimedean property as they should. For reference, the transfer is:

For every hyperreal number x, there exists a hypernatural number n such that x < n​

For reference, one typical formulation of the language is as follows. Let S₀ be the set of natural numbers. Then define

$$S_{n+1} = \mathcal{P}(S_n) \cup S_n$$​

and let $S_\omega$ be the union of all the $S_i$. Roughly speaking, $S_\omega$ contains anything you can construct set-theoretically from the natural numbers using the "power set" operation only a finite number of times.

You can then define a first-order formal theory where each element of $S_\omega$ is a constant symbol, set membership is a binary relation symbol, and every true set-theoretic statement involving elements of $S_\omega$ is taken as an axiom. This is clearly complete, so any two models are elementarily equivalent.

The neat trick is that you can choose the non-standard model so that the interpretation of $\in$ coincides with the standard set-theoretic meaning. However ${}^\star \mathcal{P}(A) \neq \mathcal{P}({}^\star A)$ whenever A is an infinite set.

Anyways, since C is an element of $S_\omega$, it has an interpretation in the non-standard model. And if in the standard model C was defined as RxR, the same must be true in the non-standard model. (execise: check that ${}^\star A \times {}^\star B = {}^\star A \mathop{ {}^\star \times} {}^\star B$)

A standard complex number can be conceptualized as a pair of real numbers (x,y) of the form "x + iy" where i is the square-root of -1. I am wondering if/how the non-standard complex numbers can be constructed.

Any of the usual ways to construct the complexes transfers, so you really do just repeat verbatim any of the standard constructions you're familiar with.


For the record, the calculus of hyperreal numbers can be defined by transfer too. For example, the statement

$$\lim_{x \to a} f(x) = L$$​

is true for non-standard a, L, and f if and only if
[iindent]$$\forall \epsilon > 0 \exists \delta > 0 : 0 < |x - a| < \deta \implies |f(x) - L| < \epsilon$$[/indent]
where $\epsilon, \delta$ are varying over positive hyperreal numbers.

It just that this is somewhat less efficient way to go about things.

Does anyone know where I could find an online book or something along those lines? I know Robinson briefly mentioned the complex extension in his book, but that's about as much as I have seen.

Unfortunately, most of what I've learned I've picked up from scattered reading, and I can't really recommend or even recall anything.

Keisler's textbook goes into multi-variable calculus; in some sense, complex analysis is a manifestation of the properties of differential forms -- and differential geometry is built upon multi-variable calculus. This might be a reasonable line to pursue?

 

[JungleJesus]

Now, "reals" and "hyperreals" are, in some sense, the simple version. The grander method of non-standard analysis is to construct a formal theory that let's you describe not only algebra, but enough of set-theory to describe analysis too.


Any of the usual ways to construct the complexes transfers, so you really do just repeat verbatim any of the standard constructions you're familiar with.


For the record, the calculus of hyperreal numbers can be defined by transfer too. For example, the statement

$$\lim_{x \to a} f(x) = L$$​

is true for non-standard a, L, and f if and only if

$$\forall \epsilon > 0 \exists \delta > 0 : 0 < |x - a| < \deta \implies |f(x) - L| < \epsilon$$​

where $\epsilon, \delta$ are varying over positive hyperreal numbers.

It just that this is somewhat less efficient way to go about things.

Cool. I'll look into this process in more detail. Who knows, I might even write the book I'm looking for. Thanks for the attention.

 

[Bacle]

Thanks, Hurkyl, very helpful.