# Intermediate Extensions Determined by Reals

Does there exist a forcing $P$ which adds a generic real in the sense that $V[G] = V[x]$ for some $x \in ({}^\omega\omega)^{V[G]}$, and for all reals $y \in ({}^\omega\omega)^{V[G]}$, if $V[y] \neq V$, then there exists some $z \in {}^\omega\omega$ such that $V \subsetneq V[z] \subsetneq V[y]$?

Is there a forcing poset $P$ which adds a generic real and in $V[G]$ there are no finite sets $\{z_1, ..., z_n\}$ such that for all reals $V[z] = V[z_i]$ for some $i$, and if $A \subseteq ({}^\omega\omega)^{V[G]}$ and $A \in V[G]$ there exists some $y \in A$ such that $V \subsetneq V[y]$ and $V[y]$ is $\subseteq$ minimal among $\{V[z] : z \in A\}$.

This question is motivated by forcing (like Sack's Forcing) which adds a minimal real degree. Adding finite real degrees means that there is a finite set $\{z_1, .., z_n\}$ such that for all $y$, $V[y] = V[z_i]$ for some $i$.

I am interested in the behavior of forcing extension that do not add finite real degrees. The two questions above can be phrased differently as follows: For the first question I want to know if there is a generic extension such that the proper intermediate extensions given by reals are not well-founded under $\subseteq$. The second question, is whether there is a extension that does not add finite real degrees whose proper intermediate extensions given by reals are well-founded in the sense above.

Thanks for any information.

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I think random forcing has the property. It can be proved that if $x$ is random and $y\in L[x]\setminus L$, then there is a random real $z\in L[y]$. –  Liang Yu May 15 at 2:23
For your second property, what about something like Sacks forcing iterated $\omega_1$-many times with countable supports? –  Noah S May 15 at 2:37
I think Noah's suggestion for the second question is right, but I don't actually know that. The key question seems to be what sorts of new reals show up at stage $\omega$ of the iteration. Note that, if Noah's $\omega_1$-stage iteration does the job, then so does the submodel obtained by iterating for only $\omega$ steps. –  Andreas Blass May 15 at 13:54
Oops (about my previous comment): The question requires that the whole forcing extension be generated by a single real. So, if a countable-support Sacks iteration works, then it should be only $\omega$ steps, not $\omega_1$. –  Andreas Blass May 15 at 13:56
Oh, I didn't notice that the extension needed to be generated by a single real. So presumably Sacks iterated $\omega$-many steps with finite supports would be a forcing to look at? –  Noah S May 19 at 22:12