Chapter 2 introduces the Solvability Complexity Index (SCI) hierarchy for classifying the difficulty of computational problems and establishing algorithmic optimality. A computational problem is defined by an input class, an output space, and the information accessible to an algorithm. We formalise algorithms and towers of algorithms whose iterated limits converge to the desired output, and define the SCI as the minimal number of nested limits required. The framework accommodates inexact data and shows that, for many operator problems, results are robust across standard computational models. We then incorporate verification—computer-assisted proof and one-sided error bounds for certified spectra—and give a quantifier-based characterisation of decision problems. Canonical combinatorial problems are linked to the Baire and Borel hierarchies. These problems are then embedded into the classical spectral problem, proving that computing the spectrum of a general bounded operator requires three successive limits. The argument pinpoints two obstructions: missing global information about the operator’s action and spectral instability for non-normal operators.