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ArisKatsaris comments on Logical Pinpointing - Less Wrong
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The Peano Arithmetic talks about the Successor function, and jazz. Did you know that the set of finite strings of a single symbol alphabet also satisfies the Peano Axioms? Did you know that in ZFC, defining the set all sets containing only other members of the parent set with lower cardinality, and then saying {} is a member obeys the Peano Axioms? Did you know that saying you have a Commutative Monoid with right division, that multiplication with something other than identity always yields a new element and that the set {1} is productive, obey the Peano Axioms? Did you know the even naturals obey the Peano Axioms? Did you know any fully ordered set with infimum, but no supremum obey the Axioms?
There is no such thing as "Numbers," only things satisfying 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:
From these definitions we get, example-wise: