There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some real number $z$. Hence my question is the following:

Question: Is there an analytic equivalence relation with Borel classes in $L$ ( or in $L[z]$ for some real number $z$ ) but non Borel classes in some other model of set theory ( or under large cardinal assumptions, or when assuming the existence of $0^{\#}$ ).

Following are the examples I know about:

Example 1: in $L$ let $A$ be an uncountable $\mathbf{\Pi_1^1}$ set with no perfect subset. Consider the analytic e.r. whose classes are $\mathbb{R}-A$ and all the singletons of $A$. Then there is a non Borel class, but after collapsing $\omega_1$ over $L$, all classes become Borel.

Example 2: Fix a $\Pi_1^1$ formula $\Phi(x,z)$ such that for every $z$: $\{x\ :\ \Phi(x,z)\}$ is an uncountable (in $L[z]$) $\mathbf{\Pi_1^1}(z)$ set with no perfect subset. Then consider the following e.r.:$(x,z)E(y,z')\iff (z=z')\wedge((\neg\Phi(x,z)\wedge\neg\Phi(y,z))\vee(x=y))$. The classes of the $z$ - section are either singletons or of size $\omega_1^{L[z]}$. Hence all classes are Borel if and only if $\forall z:\ \omega_1^{L[z]} < \omega_1$.

Notice that a non Borel class may be added by forcing the universe to be $L[z]$, but that still does not answer the question above.


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