The definitions only decorate the truths. By unwrapping the definitions, the truths can be expressed using just the barest predicates and function symbols of t

The definitions only decorate the truths. By unwrapping the definitions, the truths can be expressed using just the barest predicates and function symbols of the background language, and logical operators. Whether you define 0^0 to be 1 or not doesn't change the unwrapped truth.