I'm new in the study of the forcing method and I having some troubbles to solve some of the exercise from Kunen's book (edition 2013): specifically, problem IV 2.46 from page 271. It says the following:
"Let M a ctm for ZFC and $\mathbb{P}=Fn(\omega,2)$. Prove that there exists a filter $G$ on $\mathbb{P}$ such that there is no transitive $N\supset M$ such that $N\vDash ZF-P$, $G\in N$ and $o(M)=o(N)$."
Kunen gives as a Hint that G can be code a well-order of $\omega$ of type greater than $o(M)$. So, my questions are:
- First of all, what does mean that $G$ can be code a well-order of $\omega$?
- Secondly, how we can prove that statement?
Any help would be much appreciated.
Here's a sketch: Take $x \subseteq \omega$ coding a well order on $\omega$ of order type $> o(M)$ using the coding suggested by B. Scott. Let $G = \{p \in Fn(\omega, 2): p \subseteq x\}$. Towards a contradiction, suppose there is some transitive model $N$ of ZF - P that extends $M$, contains $G$ and has the same ordinals as $M$. Since $G \in N$, so is $\bigcup G = x$ since $N$ models union axiom. Decode $x$ to get the well ordering that it is coding. Note that the proof of the fact that every well ordered set is order isomorphic to a unique ordinal works without power set hence $N$ contains the ordinal coded by $x$. This contradicts the fact that $M$ and $N$ have the same ordinals.