uncountable cardinalities. If T is a countable complete theory, then the number I(T, ℵα) of isomorphism classes of models is given for ordinals α>0 by the minimum of 2ℵα and one of the following maps: 2ℵα. Examples: there are many examples, in particular any unclassifiable or deep theory, such as the theory of the random graph. for some countable infinite ordinal d. (For finite d see case 8.) Examples: The theory with equivalence relations Eβ for all β with β+1<d, such that every Eγ class is a union of infinitely many Eβ classes, and each E0 class is infinite. for some finite positive ordinal d. Example (for d=1): the theory of countably many independent unary predicates. for some finite positive ordinal d. for some finite positive ordinal d; for some finite positive ordinal d. Example (for d=1): the theory of countable many disjoint unary predicates. for some finite ordinal d≥2; for some finite positive ordinal d; for some finite ordinal d≥2; Examples: similar to case 2. . Example: the theory of the integers viewed as an abelian group. for finite α, and |α| for infinite α, where G is some subgroup of the symmetric group on n ≥ 2 elements. Here, we identify αn with the set of sequences of length n of elements of a set of size α. G acts on αn by permuting the sequence elements, and |αn/G| denotes the number of orbits of this action. Examples: the theory of the set ω×n acted on by the wreath product of G with all permutations of ω. . Examples: theories that are categorical in uncountable cardinals, such as the theory of algebraically closed fields in a given characteristic. . Examples: theories with a finite model, and the inconsistent theory. Moreover, all possibilities above occur as the spectrum of some countable complete theory. The number d in the list above is the depth of the theory. If T is a theory we define a new theory 2T to be the theory with an equivalence relation such that there are infinitely many equivalence classes each of which is a model of T. We also define theories by , . Then . This can be used to construct examples of theories with spectra in the list above for non-minimal values of d from examples for the minimal value of d. See also Spectrum of a sentence References C. C. Chang, H.