I am not sure where these questions belongs, beginning or advanced algebra … with regard to Peano’s axioms:

1) Must a successor function be written in a “recursive” format?

2) A successor function is certainly a one-to-one function but is that sufficient and if not is there another property that a successor function must possess?

3) Given

“a” is a real number,

“n” is a natural number and the recursive definition:

a^{1}= a, and a^{n+1}= a^{n}a, it can be proved that this recursive definition is true for all n.

Does that mean that the here stated recursive definition is itself proved to be a “successor function”? If yes, does this mean all recursive functions proved true for all “n” are successor functions?

Thanks.