Let's say we have an operator ⨁ which is recursively defined as such:
a ⊆ ℤ; b ∈ ℤ
a ⨁ b = (a ∪ {b}) ⨁ (b + 1)
This definition has no "halting case".
Intuitively, it adds b to the set a, then adds b+1 to a, and so on.
What is ∅ ⨁ 0? Is this even defined?
Intuitively (again, and probably wrongly) this should yield the set of non-negative integers, first adding 0, then 1, then 2, etc.
The operator is not well defined, The operation $a\oplus b:=c$ satisfies the property for any given set $c$.