On nonwellfounded iterated Sacks extensions, with application to the Glimm -- Effros property

Abstract

We prove that if $\bI$ is a p.\ o. set in a countable transitive model $\gM$ of $\ZFC$ then $\gM$ can be extended by a generic sequence of reals $\a_ı,$ $ı\in\bI,$ such that $\aleph_1^\gM$ is preserved and every $\a_ı$ is Sacks generic over $\gM[\ang{\a_\j:\j<ı}]$. The structure of the degrees of \dd\gM constructibility of reals in the extension is investigated. As an application, we obtain a model in which the $\is12$ equivalence relation $x \E y$ iff $\rL[x]=\rL[y]$ ($x,\,y$ are reals) does not admit a reasonable form of the Glimm -- Effros theorem.