Statements forced by one condition of a poset, but not the whole thing

Question

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be the case, most often, that the interesting statements we try to force end up being forced by the whole poset.

A sufficient property for a poset to possess to make the above phenomenon occur (in the case where all parameters in the forced statement are canonical names for objects in the ground model) is almost homogeneity: For every $p, q \in P$ there is an automorphism $i$ of $P$ such that $i(p)$ and $q$ are compatible.

It makes sense that if you're building a poset to force something, the whole poset forces it (there's also the ad hoc reason that you could throw out the part that doesn't force it). However, it might happen that in trying to force a particular statement, you build a poset where some, but not all, of the conditions happens to force a different interesting statement. Also, I haven't given this much deep thought, but it seems natural for most posets to be almost homogeneous.

My questions: Are there any interesting, instances of independence results forced by part, but not all, of some poset? Are there examples of commonly encountered posets which aren't almost homogeneous.

(If there ends up being a big list of answers, I'll add the "big-list" tag and make it community wiki)

Answer 1

There are several issues.

First, of course, if a statement $\varphi$ is forced by a condition $p\in\mathbb{P}$, then of course it is forced by every condition in the poset $\mathbb{P}|p$, which restricts $\mathbb{P}$ to conditions below $p$. So whenever a statement is forced by any condition, then there is a poset such that every condition forces the statement.

Second, there are many statements that can be forced, but not be almost homogeneous forcing. For example, one of the most important consequences of almost homogeneous forcing is that one can limit the HOD of the forcing extension. Namely, if $V\subset V[G]$ is an almost homogeneous forcing extension by $\mathbb{P}$, then the $HOD^{V[G]}\subset HOD(\mathbb{P})^V$. Thus, one can never force $V=HOD$ by nontrivial definable almost homogeneous forcing. The natural class forcing of $V=HOD$ is ordinal definable, but not almost homogeneous. One way to do it is to proceed in an Easton support class iteration, which at each stage $\gamma$, forces either the GCH to hold at $\gamma$ or to fail at $\gamma$, letting the generic filter decide which is to be done. A simple density argument (replacing the bookkeeping argument that would have been used in earlier years) ensures that every new set of ordinals is coded into the GCH pattern on a block of cardinals, thereby ensuring GCH. But this forcing is definitely not almost homogeneous, since conditions that opt to force an instance of GCH at a cardinal cannot be automorphed to those that force the opposite.

Third, when every statement is forced by every condition in a partial order $\mathbb{P}$, whether or not the partial order is homogeneous, then the theory (restricted to any level of complexity) of the forcing extension must be an element of the ground model, since we can recover it from the forcing relation. Thus, one can make an easy counterexample. Start in $L$, add a Cohen real $c$ and then code $c$ into the continuum pattern on the $\aleph_n$'s. You get a model where the GCH pattern on the $\aleph_n$'s is $L$-generic. But there can be no partial order in $L$ forcing that statement, if all statements are to be forced by every condition.

There are many other examples of non-almost- homogeneous forcing notions. For example, the generic Souslin trees added by forcing (and constructed under $\diamond$) are rigid and therefore not almost-homogeneous as forcing notions. For example, you can arrange to have a Souslin tree such that forcing with it adds exactly one branch through the tree.

Another example is the lottery preparation, which I introduced. This is like the Laver preparation, but works with a variety of different large cardinals. At stage $\gamma$, the forcing consists of the lottery sum of a family of allowed forcing notions; this is the partial order consisting of $\{1\}\cup\{(\mathbb{Q},q)\mid q\in\mathbb{Q}\in F\}$, where $F$ is the family, and the order has $1$ above everything and otherwise $(\mathbb{Q},q)\leq (\mathbb{Q}',q')\iff \mathbb{Q}=\mathbb{Q}'$ and $q\leq_{\mathbb{Q}} q'$. So the generic filter in effect chooses a winning poset from the family and forces with it. This is clearly not almost homogeneous when the family is rich. The lottery preparation is a long iteration of such lotteries, and achieves various indestructibility results.

Answer 2

Amit: A very interesting example of non-homogeneous posets are lotteries. Joel introduced them and has done a significant amount of work on them.

Given a collection $\{{\mathbb P}_i\mid i\in I\}$ of posets, their lottery is $\{(p,i)\mid p\in {\mathbb P}_i\}$ ordered by $(p,i)\le(q,j)$ iff $i=j$ and $p \le_{{\mathbb P}_i} q$.

The generic "randomly" selects a ${\mathbb P}_i$ and adds a generic for it. By starting with wildly different posets, we may end up with radically different extensions, depending on the generic.

This is particularly useful in iterations, where along the way we need to ensure a variety of posets are selected (in order to obtain, say, certain forcing axioms). It is a nice alternative to using Laver functions.

The standard reference here is J. D. Hamkins, "The lottery preparation", Ann. Pure Appl. Logic 101 (2000), 103–146. For a (sophisticated) recent application, see for example N. Dobrinen - S. Friedman, "The consistency strength of the tree property at the double successor of a measurable cardinal", Fundamenta Mathematicae 208 (2010), 123–153. (In this paper, we start with a ground model where GCH holds and $\kappa$ is weakly compact hypermeasurable, and a poset is described that preserves the measurability of $\kappa$ while forcing the tree property at $\kappa^{++}$. The poset is an Easton support iteration of a lottery of iterated Sacks forcing posets at different cardinals.)

A good example of a poset that is as far from being homogeneous as you may want is the Vopenka algebra, see Theorem 15.46 of Jech's "Set Theory". Woodin realized that Vopenka algebras are particularly useful in the study of models of determinacy, and this has been a key insight.

In the original application, Vopenka showed that if $A$ a set of ordinals, then $L[A]$ is a generic extension of its HOD. Conditions are (ordinals coding) ordinal definable sets of subsets of $\kappa$, where $A\subseteq\kappa$. The usefulness of this algebra and its variants for determinacy is that it allows us to argue about arbitrary sets of reals as if they where Borel sets. For a concrete application, see for example my recent paper with Ketchersid, "A trichotomy theorem in natural models of AD${}^+$", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.