Encoding is not definition. While it is true that we cannot decide the integer value of “1111...” as you describe, the reason behind it is obvious: you are trying to use an inductive algorithm on a co-inductive data structure.
Clearly, your proposed bits-to-ints encoding scheme is isomorphic to lists of unit, and it so happens that there is a one-to-one mapping between list-of-unit and inductively defined natural numbers, exactly so long as the lists in question are inductive and thefore finite. This isomorphism is made up of length :: list unit -> int and repeat unit :: int -> list unit, according to their usual definition. Proof of that this is isomorphic is left as an exercise to the reader.
Encoding is not definition. While it is true that we cannot decide the integer value of “1111...” as you describe, the reason behind it is obvious: you are trying to use an inductive algorithm on a co-inductive data structure.
Clearly, your proposed bits-to-ints encoding scheme is isomorphic to lists of unit, and it so happens that there is a one-to-one mapping between list-of-unit and inductively defined natural numbers, exactly so long as the lists in question are inductive and thefore finite. This isomorphism is made up of
length :: list unit -> intandrepeat unit :: int -> list unit, according to their usual definition. Proof of that this is isomorphic is left as an exercise to the reader.