The number of $L_{\infty\kappa}$-equivalent nonisomorphic models for $\kappa$ weakly compact

Saharon Shelah ¹   ; Pauli Vaisanen ²

¹ Institute of Mathematics The Hebrew University Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.
² Department of Mathematics P.O. Box 4 00014 University of Helsinki, Finland

Fundamenta Mathematicae, Tome 174 (2002) no. 2, pp. 97-126

Résumé

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.

Détail

DOI : 10.4064/fm174-2-1

Keywords: cardinal kappa model cardinality kappa denote number nonisomorphic models cardinality kappa which infty kappa equivalent prove kappa weakly compact cardinal question possible values models cardinality kappa equivalent question possible numbers equivalence classes equivalence relations which sigma definable kappa cite shva possible have generic extension where possible numbers equivalence classes sigma equivalence relations prearranged set together these results settle problem possible values models weakly compact cardinality