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ANALYTIC EQUIVALENCE RELATIONS SATISFYING HYPERARITHMETIC-IS-RECURSIVE

  • ANTONIO MONTALBÁN (a1)
Abstract

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  $ZF$ .

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
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Forum of Mathematics, Sigma
  • ISSN: -
  • EISSN: 2050-5094
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