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.
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.