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Jensen proved that given $V\models\sf ZFC+GCH$, there is a class generic real $r$, such that $V[r]=L[r]$, and no cardinals are collapsed.

We know that this can be modified such that $r$ is minimal, i.e. if $x\in L[r]$, then $x\in V$ or $r\in L[x]$. And we know that we can replace "real" with "subset of $\kappa$" for some arbitrary (regular?) cardinal.

But can do better?

  1. Can we code the universe into a "Prikry sequence"? (Preserving cardinals, but not cofinalities.)
  2. Can we code the universe into a "Namba sequence"? (Permitting whatever cardinals need to be collapsed to be collapsed, of course.)

Of course, this is a bit open-ended, and there might be related results out there which I didn't quite mention. Those are all good and well.

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    $\begingroup$ Assume in $V$ there is a measurable cardinal $\kappa$ and let $U$be a normal measure over it. Force with Prikry forcing to add an $\omega$-sequence $C=(\kappa_n: n<\omega)$ over $V$. Force over $V[C]$ by Jensen's coding to find a generic extension of the form $V[C][G]=L[r]$, for soeme real $r$. Define a new sequence $D=(\kappa_n: n \in r)$. Then $V[C][G]=V[C][D]$. The key point is that having $D, C$ we can recover $r$. $\endgroup$
    – Mohammad Golshani
    Commented Dec 1, 2020 at 14:45
  • $\begingroup$ Okay, can we do that without adding reals, then? (Yes, code the whole damn thing into a subset of $\omega_1$; by induction: can we do it without adding bounded subsets to $\kappa$?) $\endgroup$
    – Asaf Karagila ♦
    Commented Dec 1, 2020 at 15:21
  • $\begingroup$ I can prove the following: suppose $\kappa$ is an inaccessible limit of measurable cardinals and let $S$ be a discrete set of measurables below $\kappa$ of size $\kappa$. Consider the corresponding Prikry product and let $C=(C_\alpha: \alpha \in S)$ be generic for it, where each $C_\alpha$ is an $\omega$-sequence cofinal in $\alpha$. Now code everything into a subset of $\kappa$, adding no bounded subsets to $\kappa.$ We may assume the generic is some $X \subset S$. Thus $V[C][X]=L[X]$ Now one can define a new sequence $D=(D_\alpha: \alpha \in S)$ such that $V[C][X]=V[C][D]$. $\endgroup$
    – Mohammad Golshani
    Commented Dec 2, 2020 at 4:40
  • $\begingroup$ I kinda get the feeling that you're selling me the same idea as before, but now instead of a real, it's a subset of an inaccessible limit of measurables. :-) $\endgroup$
    – Asaf Karagila ♦
    Commented Dec 2, 2020 at 9:53
  • $\begingroup$ Yes, the same. The problem is that I don't see how to code a large set into a Priky sequence. Note that for example by Woodin's result, $L[C]$ satisfies CH, when $C$ is an $\omega$-sequence of ordinals. But for example, I can have $L[X]$ may fail to satisfy CH, if $X \subset \omega_2$, so in some sense, there might be some difficulty in general. $\endgroup$
    – Mohammad Golshani
    Commented Dec 2, 2020 at 10:52

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