Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not too far" from $V$?

Question

In the following, whenever I say "$V_1$ is an outer model of $V_2$", I mean

• $V_1, V_2$ are transitive models of $\mathsf{ZFC}$,
• $V_2 \subset V_1$,and
• $ORD^{V_1} = ORD^{V_2}$.

I am curious if there exists a pair of universes $(V, W)$ such that

1. $V \subset W$,
2. $V, W$ are outer models of $L$,
3. $V \neq L[X]$ for all sets $X \in V$,
4. $W = V[Y]$ for some set $Y \in W$ (that is, $W$ is the smallest outer model of $V$ containing $Y$), and
5. $W$ is not a set forcing extension of $V$.

Basically, I want $V$ to not be generated by any set over $L$, and $W$ to be generated by some non-generic set over $V$. The latter clause is not hard to satisfy if $V$ is of the form $L[X]$, but I cannot come up with such a $W$ when $V$ is so far from $L$.

Ideally, I would want $V$ to be a definable class in $W$. If that is not possible, can there be countable $V$ and $W$ with the same properties (just replace $L$ in Points 1 to 3 with $L^V = L^W$)?

Answer 1

One example: do some class forcing over $L$ to produce a proper class $V\models$ ZFC + "there is no set-forcing which forces that $V[G]=L[x]$ for a set $x$". Now do Jensen coding forcing over $V$ to produce $W=V[x]=L[x]$ with a real $x$.

A second example: class force over $L[U]$ to produce a model $W$ satisfying "there is a measurable cardinal $\mu$ and there is no set $X$ such that $V=L[X]$". Let $$j:W\to V=\mathrm{Ult}(W,D)$$ be the ultrapower map where $W\models$"$D$ is a $\mu$-complete non-principal ultrafilter over $\mu$". Then $(V,W)$ works, because (i) $V\models$"There is no set $X$ such that $V=L[X]$", since $W$ models that statement, and (ii) because $W=V[D]$. The latter can be seen by iteratively inverting $j$ definably over $V[D]$. That is, compute $(V_\alpha^W,j\upharpoonright V_\alpha^W)$ recursively on ordinals $\alpha$. For $\alpha\leq\mu$ this is trivial. Given $(V_\alpha^W,j\upharpoonright V_\alpha^W)$ and $\beta=\sup j``\alpha$, observe that $V_{\alpha+1}^W$ is just the set of all $j\upharpoonright V_\alpha^W$-preimages of elements of $V_{\beta+1}^V$. That is, $X\in V_{\alpha+1}^W$ iff there is $Y\in V_{\beta+1}^V$ such that $X=((j\upharpoonright V_\alpha^W)^{-1})``Y$. If $\alpha$ is a successor or has cofinality $\neq\mu$, as computed in $W$, or equivalently, in $V$, or equivalently, in $W[D]$, then moreover, $j(\alpha)=\beta$, and $j\upharpoonright V_{\alpha+1}^W$ is given by inverting the collapses $Y\mapsto X$ just computed. If instead $\alpha$ has cofinality $\mu$ (in any of those three models) then $j(V_{\alpha+1}^W)=\mathrm{Ult}(V_{\alpha+1}^W,D)$ and $j\upharpoonright V_{\alpha+1}^W$ is the corresponding ultrapower map; here the functions used to form the ultrapower are just those (coded) in $V_{\alpha+1}^W$ itself. The only information needed in this process that wasn't already in $V$ was the use of $D$ to form the ultrapower when $\alpha$ has cofinality $\mu$. And $W$ is not a set forcing extension of $V$ because there is a proper class of ordinals which are cardinals in $V$ but not cardinals in $W$. And here $V$ is definable from the parameter $D$ over $W$ (and we could have in fact arranged that $D$ is unique in $W$, and hence $V$ definable without parameters over $W$). (The general idea here is also used in my paper "Varsovian models $\omega$", and some of the calculations in "Varsovian models II" (joint with Sarsgyan, Schindler) and "Periodicity in the cumulative hierarchy" (joint with Goldberg).)

Answer 2

The answer is yes.

The argument uses Jensen's theorem on coding the universe with a real.

Theorem. Every model $V$ of ZFC has a class forcing extension $V[G]$ such that there is a real number $r\subset\omega$ such that $V[G]=L[r]$.

Using this theorem we can make an example of your phenomenon. Start with any model $V$ that is not $L[x]$ for any set $x$. Perform some class forcing to $V[H]$, which cannot arise by set forcing, such as by collapsing unboundedly many cardinals. Now perform the coding-the-universe forcing over $V[H]$ to get $V[H][G]=L[r]$, and call this $W$.

This has all the properties you requested. $W=V[r]$, since in fact $W=L[r]$, but $W$ is not a set-forcing extension of $V$, since $r$ codes $H$, which is not set-generic over $V$.