mathgrad

I am a mathematician I don't need examples. What is your formal definition of a mapping.

In mathematic it is normally used as a synonym for a morphism in a given category and in the category of sets this would be a function. So in our context a mapping is a function.

Look up Peano arithmetic to see how one proves 1+1 = 2 within the standard formal system.

A function is a relation for elements in a to elements in b such that for every element in a there is a unique element in b. This is vacuously true of the empty relation when a is empty.

There is a very natural definition of numbers as sets. We define 0 to be the empty set and we define the successor function by S(x) = {x} union x. Then the natural numbers are the smallest set containing 0 and closed under the successor operation. This is the standard way to define the natural numbers within ZFC set theory.

This is admittedly very formal and not how the lay person thinks of natural numbers. However, it gets to the point of that although we may lie and say 0^0 = 1 is just a convention it is in fact a theorem within the system we work.

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.