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 κ. We denote by μ(respectively \hatμ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. We prove that I(T,κ), as a function of κ> \aleph_0, is the minimum of 2^κ and one of the following functions: 1. 2^κ; 2. the constant function 1; 3. |\hatμ^n/{\sim_G}|-|(\hatμ- 1)^n/{\sim_G}| if \hatμ<ωfor some 1 = ωsome group G <= Sym(n); 4. the constant function \beth_2; 5. \beth_{d+1}(μ) for some infinite, countable ordinal d; 6. \sum_{i=1}^d Γ(i) where d is an integer greater than 0 (the depth of T) and Γ(i) is either \beth_{d-i-1}(μ^{\hatμ}) or \beth_{d-i}(μ^{σ(i)} + α(i)), where σ(i) is either 1, \aleph_0 or \beth_1, and α(i) is 0 or \beth_2; the first possibility for Γ(i) can occur only when d-i > 0.