# Is there a version of Miller forcing "guided by" an ultrafilter?

It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter added by the first poset. That is, Mathias forcing is the two-step iteration of a countably closed poset and a $\sigma$-centered poset. In a similar way, Laver forcing completely embeds into a closed $\ast$ centered poset. It’s natural to wonder whether their cousin Miller forcing (aka rational perfect set forcing) has the same property. While Laver and Mathias forcings have natural ccc versions guided by an ultrafilter, it’s not clear (to me) how to guide Miller forcing by an ultrafilter. Here is my question:

Does Miller forcing completely embed into the two-step iteration of a $\sigma$-closed poset and a $\sigma$-centered poset?

This question allows for slightly more than the vague question of the title; I suppose the first closed poset might be something other than $P(\omega)/\mathrm{fin}$.

One reason to ask this is that a "Yes" would imply that finite products of Miller, Laver, and Mathias forcing are all proper. [Why? Every poset completely embeddable into a closed $\ast$ centered poset is proper, and the class of such posets is closed under finite products.] Note that Spinas has already established the properness of finite powers of Miller & Laver forcing.

Of course, I’d be grateful for whatever references—for negative or positive results—you’re willing to provide.

The answer is no. Like Sacks forcing, Miller forcing adds a minimal degree, meaning that if $G$ is the Miller generic and $x\in M[G]$, then either $x\in M$ or $G\in M[x]$. Hence we cannot have Miller forcing $\mathbb{Q}$ equivalent to a two-step iteration $\mathbb{P}_0*\mathbb{P}_1$ since the generic for $\mathbb{P}_0$ contradicts the minimality of the degree corresponding to the generic for $\mathbb{P}_0*\mathbb{P}_1$.

The minimality of the generic added by $\mathbb{Q}$ is a consequence of Theorem 2 in "Combinatorics on ideals and forcing with trees" by Marcia Groszek. I think this is also covered in Miller's original paper, Rational perfect set forcing, though I don't have access right now.

• But this doesn't preclude Miller forcing from embedding into a countably closed * ccc forcing, right? Mar 12, 2015 at 17:54
• @ToddEisworth No, this would be true if "minimal degree" meant "minimal real degree", as it often does, but Miller and Groszek consider minimal degrees in the general sense: if $X \in M\setminus M[G]$ then $G \in M[X]$. Mar 13, 2015 at 1:04

More of an extended comment than an answer:

Sabok and Zapletal have considered Miller forcing "guided" by $J^+$ for an ideal $J$ in their paper "Forcing properties of ideals of closed sets", J. Symbolic Logic 76 (2011) no 3, 1070-1095.

Also, you may want to take a look at Blass's paper "Applications of superperfect forcing and its relatives", Lecture notes in mathematics 1401, 1989, pp. 18-40, where the analogy "Miller forcing is to Matet forcing as Laver forcing is to Mathias forcing" is discussed.

Matet forcing factors as $\sigma$-closed * $\sigma$-centered, so maybe you can prove something analogous to what we have for Laver.

• Thanks for the references. It didn't occur to me to look at Matet forcing. I'll think about it. Mar 10, 2015 at 2:05