ISOMORPHISM ON HYP
Sy‐David Friedman · Journal of Symbolic Logic · 2016
Abstract We 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.