FINE SPECTRA AND LIMIT LAWS .2. FIRST-ORDER 0-1-LAWS

Using Feferman-Vaught techniques a condition on the fine spectrum of a n admissible class of structures is found which leads to a first-order 0-1 law. The condition presented is best possible in the sense that i f it is violated then one can find an admissible class with the same f ine spectrum which does not have a first-order 0-1 law. If the conditi on is satisfied (and hence we have a first-order 0-1 law) we give a na tural model of the limit law theory; and show that the limit law theor y is decidable if the theory of the directly indecomposables is decida ble. Using asymptotic methods from the partition calculus a useful tes t is derived to show several admissible classes have a first-order 0-1 law.