The number of $L_{\infty\kappa}$-equivalent nonisomorphic models for $\kappa$ weakly compact
Volume 174 / 2002
Abstract
For a cardinal $\kappa$ and a model $M$ of cardinality $\kappa$ let ${\rm No}(M)$ denote the number of nonisomorphic models of cardinality $\kappa$ which are $L_{\infty,\kappa}$-equivalent to $M$. We prove that for $\kappa$ a weakly compact cardinal, the question of the possible values of ${\rm No}(M)$ for models $M$ of cardinality $\kappa$ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are $\Sigma^1_1$-definable over $V_\kappa$. By \cite {ShVa719} it is possible to have a generic extension where the possible numbers of equivalence classes of $\Sigma^1_1$-equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of ${\rm No}(M)$ for models of weakly compact cardinality.
