# Sh:718

- Shelah, S., & Väisänen, P. (2002).
*The number of L_{\infty\kappa}-equivalent nonisomorphic models for \kappa weakly compact*. Fund. Math.,**174**(2), 97–126. arXiv: math/9911232 DOI: 10.4064/fm174-2-1 MR: 1927234 -
Abstract:

For a cardinal \kappa and a model {\mathcal M} of cardinality \kappa let {\rm No}({\mathcal M}) denote the number of non-isomorphic models of cardinality \kappa which are L_{\infty\kappa}–equivalent to {\mathcal M}. In [Sh:133] Shelah established that when \kappa is a weakly compact cardinal and \mu \leq \kappa is a nonzero cardinal, there exists a model {\mathcal M} of cardinality \kappa with {\rm No}({\mathcal M})=\mu. We prove here that if \kappa is a weakly compact cardinal, the question of the possible values of {\rm No}({\mathcal M}) for models {\mathcal 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. In [ShVa:719] we prove that, consistent wise, the possible numbers of equivalence classes of \Sigma^1_1-equivalence relations can be completely controlled under the singular cardinal hypothesis. These results settle the problem of the possible values of {\rm No}({\mathcal M}) for models of weakly compact cardinality, provided that the singular cardinal hypothesis holds. - published version (30p)

Bib entry

@article{Sh:718, author = {Shelah, Saharon and V{\"a}is{\"a}nen, Pauli}, title = {{The number of $L_{\infty\kappa}$-equivalent nonisomorphic models for $\kappa$ weakly compact}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {174}, number = {2}, year = {2002}, pages = {97--126}, issn = {0016-2736}, mrnumber = {1927234}, mrclass = {03C55 (03C75)}, doi = {10.4064/fm174-2-1}, note = {\href{https://arxiv.org/abs/math/9911232}{arXiv: math/9911232}}, arxiv_number = {math/9911232} }