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Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?

  1. If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
  2. If $T$ is a tree on $\omega \times \kappa$ and $V[G]$ is a generic extension of $V$, then a $\Sigma^1_1(\text{p}[T])$ statement holds in $V$ if and only if the corresponding $\Sigma^1_1(\text{p}[T]^{V[G]})$ statement holds in $V[G]$.

Here a set of reals is called Suslin if it the projection of a tree on $\omega \times \kappa$ for some ordinal $\kappa$, and (from an answer by Carlo Von Schnitzel on Math.SE) we say that a set of reals $B$ is $\Sigma^1_1(A)$ iff there is a $\Sigma^1_1$ set of reals $D$ such that $$B(x) \iff \exists y\, \big(\forall n\, (y)_n\in A \wedge D(x,y)\big).$$ (I'm oversimplifying a bit here; in the special case that $A = \emptyset$ we need to define $\Sigma^1_1(A) = \Sigma^1_1$.)

It seems like these facts are often used in an informal way (e.g. "consider the tree $T'$ of attempts to build reals coding countable objects $x$, $y$, and $z$ having some property that is witnessed by a branch of $T$....")

It would be nice to have a reference for a general statement that encompasses all such arguments.

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  • $\begingroup$ Have you checked Moschovakis book? or Cabal seminar? $\endgroup$
    – 喻 良
    Apr 23, 2015 at 5:35
  • $\begingroup$ @喻良 I didn't see it in Moschovakis's book, although I may have missed it. I have not looked though all of the Cabal seminar volumes, although that should probably be my next step. I also looked in Feng–Magidor–Woodin (for an answer to the second question.) $\endgroup$ Apr 23, 2015 at 5:39
  • $\begingroup$ If $A\in \Delta^2_1$ is a set of reals then since $\Delta^2_1$ is closed under $\exists^{\mathbb{R}}$ and $\forall^{\mathbb{R}}$ then $\Sigma^1_1(A) \subseteq \Delta^2_1$. Actually for all $n<\omega$, we'll have $\Sigma^1_n(A) \subseteq \Delta^2_1$ so these sets should be Suslin. (using the definition below of $\Sigma^1_1(A)$ in Yizheng's answer or my comment) $\endgroup$ Apr 23, 2015 at 7:24
  • $\begingroup$ @CarloVonSchnitzel I'm just assuming $\mathsf{ZF}$ though, so I'm not sure why $\Delta^2_1$ sets would be Suslin. $\endgroup$ Apr 23, 2015 at 7:25
  • $\begingroup$ Ah you're right, you just said $\text{ZF}$ in your original post. $\endgroup$ Apr 23, 2015 at 7:27

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