ISOMORPHISM ON HYP
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AbstractWe show that isomorphism is not a complete ${\rm{\Sigma }}_1^1$ equivalence relation even when restricted to the hyperarithmetic reals: If E1 denotes the ${\rm{\Sigma }}_1^1$ (even ${\rm{\Delta }}_1^1$) equivalence relation of [4] then for no Hyp function f do we have xEy iff f(x) is isomorphic to f(y) for all Hyp reals x,y. As a corollary to the proof we provide for each computable limit ordinal α a hyperarithmetic reduction of ${ \equiv _\alpha }$ (elementary-equivalence for sentences of quantifier-rank less than α) on arbitrary countable structures to isomorphism on countable structures of Scott rank at most α.
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1984 ◽
Vol 36
(6)
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pp. 1067-1080
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1981 ◽
Vol 1
(4)
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pp. 431-450
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1971 ◽
Vol 8
(04)
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pp. 781-793
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