Suppose we have a first-order theory $T_C$ that includes a binary function $C$. $C$ is a bijection $\Sigma \to \mathbb{N}$ where $\Sigma$ is the alphabet of $T_C$. The function $C$ is defined as follows:
$$C(i,j) = \text{to_num}(\text{to_num}^{-1}(i) . \text{to_num}^{-1}(j))$$
$i,j$ are natural numbers, "." is a string concatenation.
We are supposed to prove that the set of sentences $\varphi$ $\{T_C \models \varphi\}$ is not expressible in the theory $T_C$. The proof must use Tarski's undefinability theorem.
I have managed to discover that $C(i,j)$ allows for addition of natural numbers if encoded this way:
$$\text{to_num}() = 0$$ $$\text{to_num}(1) = 1$$ $$\text{to_num}(11) = 2$$ $$\text{to_num}(\text{n times 1}) = n$$
$$\text{to_num}^{-1}(0) = \text{empty string}$$ $$\text{to_num}^{-1}(1) = \text{1}$$ $$\text{to_num}^{-1}(2) = \text{11}$$ $$\text{to_num}^{-1}(n) = \text{n times 1}$$
Therefore,
$$a+b = C(a,b) = \text{to_num}(\text{to_num}^{-1}(a) . \text{to_num}^{-1}(b))$$
This works, because we concatenate $a$ ones with $b$ ones, which when turned to numbers gives the result of the addition.
I know how to construct a proof of undefinability, given the theory of Peano arithmetics and its Godel's encoding (I know how to construct the contradiction). However, I first need to map my theory $T_C$ to Peano's arithmetic theory, but I am still unsure about how to do it.