abstract

Let T be a complete, firstorder theory in a finite or countable language
having infinite models. Let I(T,kappa) be the number of isomorphism types of
models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the
number of cardinals (respectively infinite cardinals) less than or equal to
\kappa. We prove that I(T,\kappa), as a function of \kappa > \aleph_0, is the
minimum of 2^{\kappa} and one of the following functions:
1. 2^{\kappa};
2. the constant function 1;
3. \hat\mu^n/{\sim_G}(\hat\mu  1)^n/{\sim_G} if \hat\mu<\omega for some
1
= \omega some group G <= Sym(n); 4. the constant function \beth_2; 5. \beth_{d+1}(\mu) for some infinite, countable ordinal d; 6. \sum_{i=1}^d \Gamma(i) where d is an integer greater than 0 (the depth of T) and \Gamma(i) is either \beth_{di1}(\mu^{\hat\mu}) or \beth_{di}(\mu^{\sigma(i)} + \alpha(i)), where \sigma(i) is either 1, \aleph_0 or \beth_1, and \alpha(i) is 0 or \beth_2; the first possibility for \Gamma(i) can occur only when di > 0.