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The uncountable spectra of countable theories Bradd Hart Ehud Hrushovski

Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T, κ) be the number of isomorphism types of models of T of cardinality κ. We denote by μ (respectively μ̂) the number of cardinals (respectively infinite cardinals) less than or equal to κ. Theorem I(T, κ), as a function of κ > א0, is the minimum of 2 κ and one of the following functions: 1. 2; 2. the constant function 1;