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The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I + ${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$ . If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$ , z exists, then there exists an I + ${\bf{\Delta }}_1^1$ set CX such that EC is a ${\bf{\Delta }}_1^1$ equivalence relation.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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