For a givien partial order, how many generic extensions might exist? In other words, for a boolean valued model class which dreams of a generic extension, how many unique generic objects exist for a given partial order (which gives rise to a boolean algebra naming such an extension)?



For a given partial order, how many generic extensions?
It is often the case that when we force with a partial order $\mathbb{P}$, then in the resulting forcing extension $V[G]$ there may be other $V$-generic filters and indeed many other $V$-generic filters $G'$, even giving rise to the same extension $V[G]=V[G']$. Thus, although from a the filter $G$ we may construct the forcing extension $V[G]$, we should not think of the of the forcing extension $V[G]$ as necessarily determining $G$; different filters can lead to the same forcing extension.


For example, if $\mathbb{P}$ has an automorphism $\pi:\mathbb{P}\to\mathbb{P}$ in $V$, then it is not difficult to see that $\pi"G$ is also $V$-generic, and furthermore these generic filters give rise to the same forcing extension $V[G]=V[\pi"G]$. Since many of the most-commonly-used forcing notions admit numerous automorphisms, it is quite common that we find ourselves in a forcing extension $V[G]$ for which there are numerous filters $G'\in V[G]$ for the same forcing notion $\mathbb{P}$ with $V[G]=V[G']$. And we can often count them in a precise way, as Stefan did in his answer. In particular, highly homogeneous forcing notions, having numerous automorphisms, will always lead to forcing extensions having many $V$-generic filters.


Meanwhile, there are also interesting cases where the forcing extension necessarily has fewer or sometimes only one generic filter for that order, which it seems is part of what you are asking about. For example, from $\Diamond$ one can construct Souslin trees with the *unique branch property*, meaning that the tree in the forcing extension has precisely one branch (and with a Souslin tree, the generic filters in any extension are precisely the branches), and one can view this property of a forcing notion as a strong kind of rigidity. This kind of situation with Souslin trees was the theme of my article with Gunter Fuchs on the [Degrees of rigidity for Souslin trees](http://arxiv.org/abs/math/0602482), where we separate various strong notions of rigidity into a hierarchy.


Another example is *self-encoding forcing*, the forcing notion that adds a subset of some cardinal $\kappa$, and then codes this set into the GCH pattern (or whatever), and then codes these new sets into the GCH pattern a bit higher up, and so on. The process catches its tail in $\omega$ many steps, and the result is that in the corresponding forcing extension $V[G]$, there is only one $V$-generic filter for the poset, and it is therefore definable there from parameter $\kappa$.


When a nontrivial forcing notion $\mathbb{P}$ necessarily leads to a forcing extension in which $\mathbb{P}$ has exactly one generic filter, then one should think of $\mathbb{P}$ as being far from homogeneous. 


