The following is from Halbeisen, page 289, on generic extensions:
This leads to one of the key features of forcing: By knowing whether a certain condition $p$ belongs to $G \subseteq P$, people living in $\mathbf V$ can figure out whether a given sentence of the forcing language is true or false in $\mathbf V [G]$. Moreover, it will turn out that people living in $\mathbf V$ are able to verify that in certain models $\mathbf V [G]$ all axioms of $ZFC$ are true.
I have three questions related to this passage.
One is: Why is it desirable to verify the truth or falsity of statements in $\mathbf V[G]$ from within $\mathbf V$? Why would it not be good enough to verify them in $\mathbf V [G]$?
Two: Do I understand correctly that people in $\mathbf V$ use $P$ names to verify that $ZFC$ holds in $\mathbf V [G]$? (the passage does not talk about this)
Three: I thought that if $\mathbf V$ satisfied $ZFC$ then so would $\mathbf V [G]$ but the passage says only "certain models $\mathbf V [G]$" satisfy all axioms of $ZFC$. Which do and which don't?
Many thanks for your help.