URM Computable Functions of One Variable is Countably Infinite

Theorem
The set $$\mathbf{U}$$ of all URM computable functions of $$1$$ variable is countably infinite.

Proof
Let $$\mathbf{U}$$ be the set of all URM computable functions.

For each $$f \in \mathbf{U}$$, let $$P_f$$ be a URM program which computes $$f$$.

Such a program is very probably not unique, so in order to be definite about it, we can pick $$P_f$$ to be the URM program with the smallest code $$\gamma \left({P_f}\right)$$.

This is possible from the Well-Ordering Principle.

Let us define the function $$h: \mathbf{U} \to \N$$ as:
 * $$h \left({f}\right) = \gamma \left({P_f}\right)$$

Since the same URM program can not compute two different functions of $$1$$ variable, it can be seen that $$h$$ is injective.

The result follows from Injection from Infinite to Countably Infinite Set‎.