[anankaie]

The problem is that the "infinitely small" is already defined, and not equal to zero: The infinitesimal. It is not so simple as defining what the result should be - one needs to make sure this is consistent with the rest of mathematics, or rather, with the mathematics generated by the rest of your axioms and definitions, without resulting in paradox.

If anything, division by zero generates the set of all numbers in your field. That is consistent with division being the operation of finding the multiplicative inverse, but inconsistent with the idea that arithmetic operations have only a single result, rather than a set - but this already exists in finding k-roots of numbers, so it is not completely unreasonable to treat some arithmetic operations as potentially resulting in sets.

Is it a useful definition? Unclear.