Geodesic
ANALYTIC EQUIVALENCE RELATIONS SATISFYING HYPERARITHMETIC-IS-RECURSIVE
Forum of Mathematics, Sigma, Tome 3 (2015)

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We prove, in $\text{ZF}+\boldsymbol{{\it\Sigma}}_{2}^{1}$-determinacy, that, for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_{E}$ we have that a real $X$ computes a member of the equivalence class if and only if ${\it\omega}_{1}^{X}\geqslant {\it\omega}_{1}^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$. 
@article{10_1017_fms_2015_5,
     author = {ANTONIO MONTALB\'AN},
     title = {ANALYTIC {EQUIVALENCE} {RELATIONS} {SATISFYING} {HYPERARITHMETIC-IS-RECURSIVE}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {3},
     year = {2015},
     doi = {10.1017/fms.2015.5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2015.5/}
}
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ANTONIO MONTALBÁN. ANALYTIC EQUIVALENCE RELATIONS SATISFYING HYPERARITHMETIC-IS-RECURSIVE. Forum of Mathematics, Sigma, Tome 3 (2015). doi: 10.1017/fms.2015.5

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