Is it possible to extend the domain and range of a function that maps from R to R to other sets?

Question

I am currently working on a project where I would like to define infintesimals that can be used in conjunction with the real numbers (similar to the hyperreals). Right now, I am working on an algebraic extension of $\mathbb{R}$ to allow for my infintesimals. I have a definition that works for all linear combinations of integer powers in my infinitesimal $\varepsilon$ (basically just a variable at this point). So for example, expressions like $\varepsilon^5$, $7\varepsilon^3$ and $73+\varepsilon$ are defined within my framework.

The issue right now is that I can't quite seem to find a good way of defining rational exponents. I have an idea where I would like to use the exponential function:

$\displaystyle \exp(x):=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$

$\implies a^x=\exp(x\ln a)$

So I would be able to make a statement such as:

$\displaystyle \sqrt\varepsilon=\exp(\frac{1}{2}\ln\varepsilon)$

The crux here is that $\exp:\mathbb{R}\to\mathbb{R}$, and I am not sure if you can extend this to allow for my infinitesimals. On the hand, you probably could strictly algebraically, since if we assume the $\exp$-function can take infinitesimal arguments, it would equal an infinite sum where each term has strictly integer powers ($\ln \varepsilon$ too could be represented as an infinite sum). On the other hand, I know too little about topology and about how to extend functions to know for sure.

Do you think such an extension of this function (or others for that matter) would be possible? What kind of qualities could my infinitesimals NOT have for this to be possible? Or put differently: what criteria do they need to fulfil in order for such an extension to be possible? How could you make such an extension?

Answer 1

The possibility of extending the domain of functions to a larger domain in such a way that the relevant properties of the functions are preserved is called the transfer principle. Of course, the transfer principle does not apply to just any extension of $\mathbb R$. It does apply to the field of hyperreals $\mathbb R^{\ast}$ which properly extends $\mathbb R$ and is also an ordered field. In particular, the exponential function $e^x$ extends, so that now it is defined for any hyperreal input $x$ (including infinitesimal and infinite numbers).

If you stick to analytic functions, it may be sufficient to use the smaller Levi-Civita field.

Answer 2

As noted by Mikhail Katz, the transfer principle of nonstandard analysis provides a way of transferring (certain) properties from the standard to the nonstandard (which includes the infinitesimals) objects and back.

As a partial answer to your question: the transfer principle can be non-constructive in the following ways. In other words, having this principle means that things will be `non-computable' in a well-defined way.

1. classically, the transfer principle for a certain formula class corresponds to comprehension for said formula class.

2. classically, the transfer principle for sentences, i.e. formulas without parameters, does not yield extra comprehension (or anything).

3. using intuitionistic logic, the transfer principle for fairly small formulas classes implies the law of excluded middle.

What should one do in light of 1)-3), i.e. how does one develop the infinitesimal calculus in a constructive/computable manner? One should start with the nonstandard definitions of continuity, differentiability, ... formulated in terms of $\approx$. That way, one does not need the transfer principle and everything remains fairly computable/constructive. See here for details.