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, κ) be the number of isomorphism types of models of T of cardinality κ.We denote by µ (respectively μ) the number of cardinals (respectively infinite cardinals) less than or equal to κ.Theorem.I(T, κ), as a function of κ > ℵ 0 , is the minimum of 2 κ and one of the following functions:4. the constant function 2 ;5. d+1 (µ) for some infinite, countable ordinal d;6. d i=1 Γ(i) where d is an integer greater than 0 (the depth of T ) andwhere σ(i) is either 1, ℵ 0 or 1 , and α(i) is 0 or 2 ; the first possibility for Γ(i) can occur only when di > 0.The cases (2), (3) of functions taking a finite value were dealt with by Morley and Lachlan.Shelah showed (1) holds unless a certain structure theory (superstability and extensions) is valid.He also characterized (4) and ( 5) and