Did you know that the set of finite strings of a single symbol alphabet also satisfies the Peano Axioms?
Surely the set of finite strings in an alphabet of no-matter-how-many-symbols satisfies the Peano axioms?
e.g. using the English alphabet (with A=0, B=S(A), C=S(B)....AA=S(Z), AB=S(AA), etc would make a base-26 system).
Single symbol alphabet is more interesting, (empty string = 0, sucessor function = append another symbol) the system you describe is more succinctly described using a concatenation operator:
0 = 0, 1 = S0, 2 = S1 … 9 = S8.
For All b in {0,1,2,3,4,5,6,7,8,9}, a in N: ab = a x S9 + b
Surely the set of finite strings in an alphabet of no-matter-how-many-symbols satisfies the Peano axioms? e.g. using the English alphabet (with A=0, B=S(A), C=S(B)....AA=S(Z), AB=S(AA), etc would make a base-26 system).
Single symbol alphabet is more interesting, (empty string = 0, sucessor function = append another symbol) the system you describe is more succinctly described using a concatenation operator:
0 = 0, 1 = S0, 2 = S1 … 9 = S8.
For All b in {0,1,2,3,4,5,6,7,8,9}, a in N: ab = a x S9 + b
From these definitions we get, example-wise:
10 = 1 x S9 + 0 = SSSSSSSSSS0