TITLE: Formal Definition of Finite Conditions
QUESTION [5 upvotes]: Forcing with finite conditions is a common concept used by set theorists. I was thinking about its meaning, but I couldn't find any exact definition of it. At the first glance it seemed to me that maybe by finite conditions, experts mean that the conditions of a forcing notion are finite however one can naturally turn any arbitrary forcing to another one which are isomorphic as partial orders and the conditions of former are finite. On the other hand, the conditions of many forcings with finite conditions are  intrinsically infinite.
By the way implicitly, forcing with finite conditions means every condition has finitely many information about generic object(from forcing point of view). My question is:


Is there any formal definition of forcing with finite condition? Or can some one give a formal definition of forcing with finite condition including well-known forcings with finite conditions?

REPLY [4 votes]: My impression is that this term is used loosely, without formal definition, to refer to the common situation where we have a forcing notion consisting of a partial order whose elements are all finite functions (that is, each on a finite domain) and the forcing order is functional extension. The most canonical example is Cohen forcing $\text{Add}(\omega,1)$ to add a Cohen real or actually any number of Cohen reals. Other examples include the forcing to collapse a given cardinal to $\omega$ or the forcing to add a club set using finite conditions (not with countable conditions, which is another common way to do it). In some cases, there are examples where conditions are augmented with some other finite amount of information that still counts as forcing with finite conditions even though the order isn't literally functional extension.
Every forcing notion is equivalent, of course, to a forcing notion whose conditions are all finite, since we could replace any element $p$ in a partial order with $\{p\}$, which is a singleton set and hence finite, and then redefine the corresponding order on these singletons. So every condition in the new partial order is finite. But this literal interpretation is never what is meant by the phrase, "forcing with finite conditions," and I have never seen a formal definition of the concept going significantly beyond the loose understanding given in the previous paragraph.