Terminology in forcing In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q \subseteq p$. Also, one calls a subset $D \subseteq P$ dense if $\forall p \in P \exists d \in D (d \leq p)$.
This contradicts my intuition. Shouldn't $p \leq q$ mean that $q$ is larger than $p$, or that $q$ carries more information than $p$? Why do  texts about forcing turn this around? While reading, I always have to rectify this notation in order to understand the meaning. I also wonder what density has to do with the usual notions of density.
Of course, all this is just a question of terminology, since every poset $(P,\leq)$ has a dual poset $(P,\geq)$ etc.
 A: There are two conventions here. Shelah and his collaborators (for example) use the version you indicate as "more natural": That if $p\le q$, then $q$ is a stronger condition, carrying more information.
Many people in the States (myself included) use the other convention. The reason is that, by passing to the Boolean completion of the poset, indeed $p\le q$ corresponds now to $p$ carrying more information. This is explained carefully in a few places. Kunen's Set theory book (1st edn.) covers this in Chapter II, section 3, particularly Lemma 3.3. 
Some people try to avoid the issue (Foreman does this in some papers), considering only separative posets, and then writing $p\Vdash q$ for "$p$ carries more information than $q$".
Lemma 3.3 in Kunen's book, mentioned above, is also a reference for the natural topology on a poset $\mathbb P$. Here, a basic open set is an $N_p$ where, for $p\in\mathbb P$, we set $N_p=\{q\in\mathbb P\mid q$ is stronger than $p\}$. The term "density", that you ask about, is just topological density with respect to this topology. 
Again using this topology, the Boolean completion $\mathcal B$ of $\mathbb P$ is obtained by considering the collection of regular open subsets of $\mathbb P$. Recall that an open set is regular iff it coincides with the interior of its closure. The ordering in $\mathcal B$ is simply containment: $A\le B$ iff $A\subseteq B$. The Boolean algebra operations are then defined as usual in terms of this ordering. This is indeed a complete Boolean algebra. The map $i$ sending $p$ to the interior of the closure of $N_p$ is a function from  $\mathbb P$ into $\mathcal B\setminus\{0\}$ with dense range. If $p$ carries more information than $q$, then $i(p)\le i(q)$, and if $p\perp q$ then $i(p)\cdot i(q)=0$. Forcing with $\mathbb P$ is equivalent to forcing with $\mathcal B$, and some authors prefer to think of forcing in terms of Boolean-valued models, which favors the second convention.
