# Reals which must, can't or might be added by forcing

Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the characterization of random reals as the set of reals in every full measure set coded by $W$. There are also theorems characterizing reals which cannot be added by forcing, for example the well known result that $0^\sharp$ cannot be added by forcing. What I want to know is if there are more general theorems characterizing (combinatorial, measure theoretic etc) properties of reals in $V$ guaranteeing that there is or is not a $\mathbb P \in W$ such that $x \in W^\mathbb P$?

I am particularly interested in this question with regards to properties related to cardinal characteristics of continuum. For instance, if there is a real $d \in V$ is dominating over the reals in $W$, then is there a nice way to characterize when $W[d]$ is (or is not) a generic extension of $W$?

• (1) In the characterization of random reals, you meant "full measure", not "positive measure". (2) A real that is dominating over $W$ might also encode $0^{\#}$ and therefore be unattainable by forcing. Sep 24, 2017 at 20:46
• @Andreas Blass Thanks for your comment. I have edited accordingly. 1) Yes of course you are right, I don't know why I wrote positive. Thanks. 2) I see this now, and also I think I see the problem with the "proof" that I had. Is it possible though that if there is a real dominating over $W$ there is another one dominating over $W$ which is the generic for some forcing? Or can it be the case that all dominating reals encode $0^\sharp$ or whatever else might prevent something from being addable by forcing? Sep 24, 2017 at 22:28
• If you have $0^\#$, then you also have lots of reals which are generic over $L$ for various forcings (including Laver, Hechler, and Mathias, all of which add a dominating real), because these forcings have only few subsets which are in $L$. Sep 25, 2017 at 4:23

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See

MR0332477 (48 #10804). Characterization of generic extensions of models of set theory, Fundamenta Mathematica 83 (1973), pp. 35–46.

What Bukovský does is to find a characterization for when $V$ is a $\lambda$-cc generic extension of the given inner model $W$ (for a given regular cardinal $\lambda$). Ralf actually has a nice write-up of Bukovský's theorem (organized differently from the presentation in his note with Fabiana) in his recent paper

The long extender algebra, preprint. To appear in a special issue of Archive for Mathematical Logic.

In fact, it is enough that $W$ uniformly $\lambda$-covers $V$, meaning that for all functions $f\in V$ whose domain is in $W$ and whose range is contained in $W$ there is some function $g\in W$ with the same domain and such that $f(x) \in g(x)$ and $|g(x)| < \lambda$ for all $x \in \operatorname{dom}(f)$.

The generality of the result may be a bit of a drawback for the question at hand, however, since it does not seem to provide us with a widely applicable combinatorial characterization allowing us to identify those reals that are generic over $W$.

The same comment applies to the version of the result described in the note by Ralf and Fabiana. Essentially, given $r$, they look at $W=L$ and $V=L[r]$, and describe whether $W$ uniformly $\lambda$-covers $V$ in terms of possible liftings of elementary embeddings $j\!:L_\alpha\to L_\beta$. The connection with Bukovský's characterization is made explicit in page 7 of their note.

• Sorry I forgot to accept this answer a while back but it was very helpful even though it didn't completely answer my question. I've accepted it now. Thanks for you help. Dec 7, 2017 at 21:32

The paper When is a given real generic over L? by Fabiana Castiblanco and Ralf Schindler characterizes all reals which are generic over $L$.