The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham:



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> Suppose that $L[A], L[B]$ have no non-constructible reals and that $\aleph\_1^{L[A,B]}=\aleph\_1^L.$ Then $L[A,B]$ has no non-constructible reals.
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How can we prove this result.



Producing no non-constructible reals
(I don't know if this is Abraham's reasoning.) Let us write $\omega\_1$ for $\omega\_1^L$ throughout. Suppose for a contradiction the assumptions hold but that $L[A,B]$ has nonconstructible reals.


1) First note that the answer is easy if both $A,B \subseteq \omega\_1$:for then for some $\alpha < \omega\_1$ $L\_\alpha[A\cap\alpha, B\cap \alpha]$ has a non-constructible real by Lowenheim Skolem. But the latter structure is in $L$, which is absurd.


2) Otherwise: first generically (possibly class-generically) extend $L[A,B]$ to a GCH model $L[A,B][G]$ followed by a Jensen style coding to code $A,B,G$ by some $H\subseteq \omega\_1$: we then have $L[A,B,G][H]\models
$ ``$V=L[H]$''. Neither of these forcings add reals (being $\omega$-distributive). However by our assumptions $L[H]$ has non-constructible reals; but now we are in the situation of part 1) and get a contradiction as before.


