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The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$.

Fact 3.1 For E and F Borel equivalence relations one has $$E \leq_{L(\mathbb R)} F$$ if and only if $$|\mathbb R/E|_{L(\mathbb R)} \leq |\mathbb R /F|_{L(\mathbb R)}.$$

My understanding is that the first inequality is a reduction via a function in ${L(\mathbb R)}$, and the second inequality is a comparison of cardinalities within ${L(\mathbb R)}$. What is the proof of this result? I am particularly interested in whether this result holds in the Chang model, assuming convenient large cardinal axioms.

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  • $\begingroup$ Isn't it ever so slightly easier to say "work in $L(\Bbb R)$ ..." and then just omit the $L(\Bbb R)$ and all the continuing references to which model these things are computed in? $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 27, 2016 at 6:59
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    $\begingroup$ I think the first reducibility should be Borel reducibility. Otherwise it would be a simple tautology. $\endgroup$
    – Yizheng Zhu
    Commented Mar 27, 2016 at 7:12
  • $\begingroup$ @YizhengZhu: I think that the statement is not a simple tautology, due to choice issues. $\endgroup$
    – Andre Kornell
    Commented Mar 27, 2016 at 7:45
  • $\begingroup$ @AsafKaragila: I copied the statement exactly as it appears in the slides, because I am not completely confident of the meaning of this notation. $\endgroup$
    – Andre Kornell
    Commented Mar 27, 2016 at 7:47
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    $\begingroup$ The statement has a typo. He has not defined $\le_{L (\mathbb R)} $. The second inequality is awkwardly written, but it is his notational convention to indicate that there is an injection in $L (\mathbb R) $ from one quotient to the other (but this is the standard definition of $\le_{L (\mathbb R)} $). Presumably he meant to write that $\mathsf{AD} $ proves that there is an injection between the quotients constructible from $\mathbb R $ iff there is Borel injection. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 27, 2016 at 14:05

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