It really boils down to the context and the problem, any recursive definition could potentially be infinite recursive. Our best bet is always keeping the domain in check for the said problem and evaluating our proof. Also, even in formal math proofs a lot of things are just skipped over and considered a given. Division by zero is one of those.
Though, integer representation is a pretty fluid problem, in that there can be more than one representations that get the job done. So, when using lambda calculus or purely functional programming as a basis for proof one could argue that there may exist a better representation.
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