Is forcing force on us

[Timothy Y. Chow]

I thought of a related question that might be relevant.

It is a standard theorem that if M is a c.t.m. of ZFC, then M[G] is the 
smallest c.t.m. of ZFC containing both M and G (meaning that if N is a 
c.t.m. containing both M and G then N contains M[G]).  But if I understand 
the proof correctly, M[G] is also the smallest c.t.m. of ZF containing 
both M and G, and (I think) the smallest c.t.m. of ZF minus P containing 
both M and G.  How much do we have to weaken the axioms so that M[G] is no 
longer the smallest c.t.m. of the weakened axioms to contain both M and G?

Tim