# Sh:719

• Shelah, S., & Väisänen, P. (2002). On equivalence relations second order definable over H(\kappa). Fund. Math., 174(1), 1–21.
• Abstract:
Let \kappa be an uncountable regular cardinal. Call an equivalence relation on functions from \kappa into 2 \Sigma_1^1-definable over H(\kappa) if there is a first order sentence \phi and a parameter R\subseteq H(\kappa) such that functions f ,g \in {}^\kappa 2 are equivalent iff for some h\in {}^\kappa 2, the structure (H(\kappa),\in,R,f,g,h) satisfies \phi, where \in, R, f, g, and h are interpretations of the symbols appearing in \phi. All the values \mu, 1\leq\mu \leq\kappa^+ or \mu=2^\kappa, are possible numbers of equivalence classes for such a \Sigma_1^1-equivalence relation. Additionally, the possibilities are closed under unions of \leq\kappa-many cardinals and products of <\kappa-many cardinals. We prove that, consistent wise, these are the only restrictions under the singular cardinal hypothesis. The result is that the possible numbers of equivalence classes of \Sigma_1^1-equivalence relations might consistent wise be exactly those cardinals which are in a prearranged set, provided that the singular cardinal hypothesis holds and that some necessary conditions are fulfilled.
• published version (21p)
Bib entry
@article{Sh:719,
author = {Shelah, Saharon and V{\"a}is{\"a}nen, Pauli},
title = {{On equivalence relations second order definable over $H(\kappa)$}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {174},
number = {1},
year = {2002},
pages = {1--21},
issn = {0016-2736},
mrnumber = {1925484},
mrclass = {03E35 (03C55 03C75)},
doi = {10.4064/fm174-1-1},
note = {\href{https://arxiv.org/abs/math/9911231}{arXiv: math/9911231}},
arxiv_number = {math/9911231}
}