Fine Spectra and Limit Laws Ii. First-order 0{1 Laws

Preliminary Version Using Feferman-Vaught techniques a condition on the ne spectrum of an admissible class of structures is found which leads to a rst-order 0{1 law. The condition presented is best possible in the sense that if it is violated then one can nd an admissible class with the same ne spectrum which does not have a rst-order 0{1 law. If the condition is satissed (and hence we have a rst-order 0{1 law) we give a natural model of the limit law theory; and show that that the limit law theory is decidable if the theory of the directly indecomposables is decidable. Using asymptotic methods from the partition calculus a useful test is derived to show several admissible classes have a rst-order 0{1 law. 1 Front-loaded classes We will continue using the notation of Part I, the rst paper 1] of this sequel. First we study, in an abstract setting, the key property of ne spectra which suuces to prove 0{1 laws exist. In this section a subscripted lower case letter is used for members of a sequence, e.g. (a n), and the corresponding upper case letter for the partial sum function, e.g. A(x) = P nx a n .