Approximating Approximate Reasoning: Fuzzy Sets and the Ershov Hierarchy

Computability theorists have introduced multiple hierarchies to measure the complexity of sets natural numbers. The Kleene Hierarchy classifies according first-order their defining formulas. Ershov $\Delta^0_2$ with respect number mistakes that are needed approximate them. Biacino and Gerla extended realm fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., real interval $[0; 1]_\mathbb{R}$). In this paper, we combine set theory, by introducing investigating Fuzzy Hierarchy. particular, focus on $n$-c.e. which form finite levels hierarchy. Intuitively, is if its function can be approximated changing monotonicity at most $n-1$ times. We prove does not collapse; that, analogy classical case, each represented as Boolean combination c.e. sets; but contrary exhaust class all sets.