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.