Say that we have a countable support forcing iteration $ \mathbb{ P}_{ \alpha }$ ( using Jech's definition ) where $ \alpha $ is a limit ordinal, and consider two conditions $ f , g \in \mathbb{ P}_{ \alpha }$ .
I am wondering if to show that $f \sim g $ ( compatible in $ \mathbb{ P}_{ \alpha }$), it is enough to show that :
$$ \forall \beta < \alpha ~~f \upharpoonright \beta \sim g \upharpoonright \beta ~( \text{ in } \mathbb{ P}_{ \beta }).$$
( Where by $ \sim $ we mean compatible in the poset)
Should I simply construct a condition that witnesses what we want? Thanks
Unfortunately, I'm not familiar with Jech's notion of iterated forcings and hence there is some communication barrier between the two of us. I usually think of iterated forcings in terms of complete Boolean algebras and substructures of inverse limits, but these notions are compatible in the sense that each one may be translated into to other in a fashion that yields equivalent forcings.
The answer to your question is no, but - since I'm unfamiliar with your notation - I don't know of an easy way to provide you with a counterexample. If you'd like to look at iterated forcings in terms of complete Boolean algebras, the paper A Boolean Algebraic Approach to Semiproper Iterations by Vilae, Audrito and Steila contains a counterexample in Proposition 5.4.