# Forcing notions adding minimal reals

I am looking for a comprehensive list of known forcing notions which add a minimal real into the ground model. I know some of them like the Sacks forcing, or the Judah-Shelah's example of a c.c.c. forcing which adds a minimal real (and a few more).

Question. What are other known (or less known) examples of forcing notions which add a minimal generic real?

• A close relative of Sacks reals are Jensen reals, which in addition to being minimal, are $\Delta^1_3$-definable. They were introduced in Jensen' paper: Definable sets of minimal degree. Mathematical logic and foundations of set theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 122–128. North-Holland, Amsterdam, 1970. – Ali Enayat Oct 12 at 18:44

Splitting forcing is a little-known forcing that adds a splitting real and creates a minimal extension. It consists of splitting trees ordered by inclusion. A splitting tree is a perfect tree $$T \subseteq 2^{<\omega}$$ where for every $$s \in T$$ there is $$m \in \omega$$ so that for all $$n \geq m$$, $$i \in 2$$, there is $$t \in T$$, $$s \subseteq t$$ with $$t(n)= i$$.

In my recent preprint (arxiv link), I show that splitting forcing adds a minimal real (Corollary 4.20) and provide part of an argument for the minimality of the forcing extension (Corrolary 3.21).

Other than that, splitting forcing is $$\omega^\omega$$-bounding but as far as I know it is unknown whether it has the Sacks-property (see also Question 6.4 in On splitting trees by Laguzzi, Mildenberger and Stuber-Rousselle).

• Thanks a lot for the answer. – Mohammad Golshani Oct 18 at 5:43

Prikry-Silver forcing adds a minimal real: that is, if $$g$$ is Prikry-Silver generic over $$V$$ and $$h$$ is any real in $$V[g] \setminus V$$, then $$V[g] = V[h]$$.

Interestingly, while Prikry-Silver forcing is minimal for reals, it is not fully minimal (like Sacks forcing). If $$g$$ is Prikry-Silver generic over $$V$$, then there is a set $$F \in V[g] \setminus V$$ such that $$V[F] \neq V[g]$$. (Specifically, this is true when $$F$$ is the filter generated by all complements of domains of conditions in the generic; or, equivalently, if $$g$$ is the Prikry-Silver generic real, then $$F$$ is the filter generated by $$\{ A \in V \cap \mathcal P(\omega) : g \restriction A \notin V \}$$. Then $$F$$ is a new non-principal ultrafilter on $$\omega$$, and $$V[F]$$ contains no new reals.)

Edit: The minimality-for-reals of Prikry-Silver forcing can be seen as a special case of the following theorem of Grigorieff:

If $$\mathcal U$$ is a selective ultrafilter, then (what's now called) the Grigorieff forcing on $$\mathcal U$$ adds a minimal real. The conditions in this forcing are the partial functions $$A \rightarrow 2$$ where $$A \notin \mathcal U$$, and the ordering is the natural one (extension).

The reason this implies Prikry-Silver forcing adds a minimal real (but is not fully minimal) is that Prikry-Silver forcing can be factored as a $$2$$-step iteration: first force with $$\mathcal P(\omega)/\mathrm{fin}$$ to add a selective ultrafilter, and then force with the Grigorieff forcing on that ultrafilter. The first step adds no new reals, and the second step adds a minimal real by the theorem quoted above.

(All of this stuff can be found in this paper of Grigorieff. This old MO thread is also relevant.)

• Can you see Prikry–Silver as an iteration of adding a new ultrafilter, then adding a real splitting it somehow or something? – Asaf Karagila Oct 12 at 14:49
• @AsafKaragila: Yes, I think. I'll need to check to be sure, but I think that adding a Prikry-Silver real is equivalent to: (1) forcing with $\mathcal P(\omega) / \mathrm{fin}$ to add a new ultrafilter, and then (2) adding a new real using the Grigorieff forcing associated to the new ultrafilter. – Will Brian Oct 12 at 15:00
• @WillBrian Thanks Will for the answer. May you also give a reference. – Mohammad Golshani Oct 18 at 5:42
• @MohammadGolshani: See this paper of Grigorieff -- citeseerx.ist.psu.edu/viewdoc/…. – Will Brian Oct 20 at 14:06