For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are -equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ¹₁-definable over . By [SV] it is possible to have a generic extension where the possible numbers of equivalence classes of Σ¹₁-equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of No(M) for models of weakly compact cardinality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-2-1,
author = {Saharon Shelah and Pauli Vaisanen},
title = {The number of $L\_{$\infty$$\kappa$}$-equivalent nonisomorphic models for $\kappa$ weakly compact},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {97-126},
zbl = {0998.03030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-2-1}
}
Saharon Shelah; Pauli Vaisanen. The number of $L_{∞κ}$-equivalent nonisomorphic models for κ weakly compact. Fundamenta Mathematicae, Tome 173 (2002) pp. 97-126. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-2-1/