Defining positional numeral systems without binary arithmetic operations

Question

Given a totally ordered set of digits $\mathcal{D}$ with cardinality $b\in\mathbb{N}$, least element $d_{\operatorname{min}}\in\mathcal{D}$ and greatest element $d_{\operatorname{max}}\in\mathcal{D}$, there exists a unique monotone function $f:\mathcal{D}\rightarrow\mathbb{N}$ such that $f(d_{\operatorname{min}}) = 0$ and $S(f(d_{\operatorname{max}})) = b$, where $S$ is the successor function in the Peano axioms for the natural numbers. A natural number in a base $b$ positional numeral system is typically defined as a finite sequence of digits $d_{n} \ldots d_{S(0)} d_0 $ where $$d_{n} \ldots d_{S(0)} d_0 = \sum_{i=0}^{n}f(d_i) b^i$$ This definition requires addition, multiplication, and exponentiation. Are there any definitions of a base $b$ positional numeral system that do not require defining the three binary arithmetic operations but still satisfies the Peano axioms?

Answer 1

The Peano axioms do not mention anything about representations of numbers. You can define the successor just in terms of digit strings by incrementing the last digit unless it carries and so on. This gives a translation of digit strings to the unary representation as you just count how many times you need to apply successor to get a given string.

Answer 2

The set of all finite strings of your given set $\mathcal{D}$ is given by the Kleene closure $\mathcal{D}^*$, with $\mathcal{D}\subset\mathcal{D}^*$. Since you have already defined a monotonic function $f:\mathcal{D}\rightarrow\mathbb{N}$ representing the natural numbers less than $b$, we shall use that function: we can define zero $0\in\mathcal{D}^*$ and the successor function $S:\mathcal{D}^*\rightarrow\mathcal{D}^*$ to be the following:

• $0=\epsilon$, the empty string.
• For a string $a\in\mathcal{D}^*, a = d_{\operatorname{min}}a$.
• For a one-digit string $d\in\mathcal{D}, d\neq d_{\operatorname{max}},S(f(d)) = f(S(d))$.
• For a string $b = ad, a,b\in\mathcal{D}^*, d\in\mathcal{D}, d\neq d_{\operatorname{max}},S(b) = aS(d)$.
• For a string $b = ad_{\operatorname{max}}, a,b\in\mathcal{D}^*, S(b) = S(a)d_{\operatorname{min}}$.

This defines your base $b$ positional numeral system satisfying the Peano axioms.