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OK, I'll try to sort through your latest post here. Let me know if I misinterpret something.
First, you specify a theory (axiomatic system), namely ZFC-P+""P(P(omega)) exists"". Let's call that T0 for easy reference.
Now you argue that T0 proves the existence of a model (let's call it M1  the 1 because it's not a model of T0, but of a new theory), such that:
M1 satisfies ZFC-P
P(omega) is in M1. Now, do you mean the real P(omega), or do you mean that M1 satisfies ""P(omega) exists""  that is, that there's an element of M1 that M1 thinks is P(omega)?  I'm guessing the latter  that M1 satisfies ""P(omega) exists"".
You say ""aleph2 is in the model and is less than or equal to [the cardinality of] P(omega)"". Again I'm guessing you mean that this is in the model's opinion; that is, M1 satisfies the statement "" exists and |"".
OK, so going with my guesses thus far, I agree that T0 proves the existence of such an M1. M1 might be, for example, M0[G], where M0 equals  (which is a model for T0), and G is M0-generic for the partial order to add  Cohen reals.
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