We know that we can't define division by zero "in any mathematical system that obeys the axioms of a field", because it would be inconsistent with such axioms.
(1) Why can we define $a^0$ ($a\neq 0$) to be $1$? Is it possible to prove that such definition is consistent with any rule of arithmetic? How to conclude that to define $a^0$ ($a\neq 0$) we don't need abolish any other basic rule of arithmetic?
(2) More generally, how to know if a definition is consistent with a given mathematical theory?