A definable hyperreal system Reading about hyperreals I learned that a serious problem with such systems is their undefinability.
So I tried to construct a definable system of hyperreals by introducing an infinitismal element $\varepsilon$ such that
$$0^\varepsilon=\frac{1}e$$
by definition.
It seems to me that such system would inevitably include divisors of zero but I am not sure.
I wonder whether such system viable and whether it in fact includes divisors of zero.
 A: I would like to challenge a premise of the original question, namely the assumption that the hyperreal number system is "undefinable".  Most researchers in the field expected this to be the case, but Kanovei and Shelah proved in 2003 the existence of a definable model (see here).
Note. In this recent article (to appear in Journal of Symbolic Logic) we study minimal set-theoretic conditions that enable the construction of a definable hyperreal line.
A: The real numbers $\mathbb R$ is(usually thought of as) a field which means it's got operations 0,1,+,-,*,/ with some conditions (associativity, distributivity, etc.)
But you can also view $\mathbb R$ as a ring by throwing away division: that means it's just got operations +,-,*.
When you do this, it's possible to add an element $e$ satisfying $e^2 = 0$ (so $e$ is a zero divisor) to get a new ring $\mathbb R[e]/(e^2).$
Inside this ring you can define $D(f) = f(x+e) - f(x)$ and find e.g. $D(x^2) = x^2 + 2xe + e^2 - x^2 = 2xe$ which is the derivative of $f$ times $e$.
The problem is this isn't a field so you can't do division.. it's much harder to build a field with infinitesimals in it but it can be done using techniques from logic called nonstandard arithmetic.
A: A hyperreal system with the Transfer Principle (as required in nonstandard analysis) is, indeed, not definable.
But other hyperreal systems are definiable.  By "hyperreal system" I mean an ordered field that properly contains the real numbers.  In particular, these fields have infinite and infinitesimal elements.  
Some of the best known of these are:
$\bullet$ rational functions $\mathbb R(X)$ with an ordering where $X > n$ for all $n \in \mathbb N$.
$\bullet$ Levy-Civita numbers
$\bullet$ Hardy fields
$\bullet$ surreal numbers
$\bullet$ transseries
