On equivalence relations Sigma_1^1-definable over H(kappa)
release_bezn222xm5cbvfzkjjmhunul5y
by
Saharon Shelah, Pauli Väisänen
1999
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 F and a parameter R subseteq H(kappa) such that functions
f,g:kappa --> 2 are equivalent iff for some h:kappa --> 2, the structure
(H(kappa),in,R,f,g,h) satisfies F, where in, R, f, g, and h are interpretations
of the symbols appearing in F. 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 <=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.
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