The uncountable spectra of countable theories

Abstract

Let T be a complete, first-order 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<n<\omega and \hat\mu if \hat\mu >= \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_{d-i-1}(\mu^{\hat\mu}) or \beth_{d-i}(\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 d-i > 0.