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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 $z\in L[y]$. – 喻 良 May 15 '14 at 2:23
For your second property, what about something like Sacks forcing iterated $\omega_1$-many times with countable supports? – Noah Schweber May 15 '14 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 '14 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 '14 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 Schweber May 19 '14 at 22:12
up vote 5 down vote accepted

I think the forcing that adds one Cohen real fits the requirements in your first question. Although it's not the case that all the new reals in the extension are Cohen reals, it is true that the submodel generated by any new real is also obtainable by adjoining a single Cohen real. So there will be strictly smaller extensions obtainable by adding, say, the restriction of that generator to the even natural numbers. And, for similar reasons, there will be continuum many reals generating distinct intermediate models.

What I wrote here about Cohen reals also works with random reals.

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