This paper draws close connections between the ease of presenting a gi
ven complexity class C and the position of the index sets I-C = {i : L
(M(i)) is an element of C} and J(C) = {i : M(i) is total boolean AND L
(M(i)) is not an element of C} in the arithmetical hierarchy. For virt
ually all classes C studied in the literature, the lowest levels attai
nable are I-C is an element of Sigma(3)(0) and J(c) is an element of P
i(2)(0); the first holds iff C is Delta(2)(0)-presentable, and the sec
ond iff C is recursively presentable. A general kind of priority argum
ent is formulated, and it is shown that every property enforcible by i
t is not recursively presentable. It follows that the classes of P-imm
une and P-biimmune languages in exponential time are not recursively p
resentable. It is shown that for all C with I-C is not an element of S
igma(3)(0), ''many'' members of C do not provably (in true Pi(2)-arith
metic) belong to C. A class H is exhibited such that I-H is an element
of Sigma(3)(0) is open, and I-H is not an element of Sigma(3)(0) impl
ies that the polynomial hierarchy is infinite.