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Hi!

I started to read the chapter 31 in Jechs book about proper forcing. Unfortunately it is written in a rather sketchy way and I do have some issues in proving a lemma about two equivalent definitions of $(M,P)$-genericity. Jech gives the following definition:

Let $(P,<)$ be a fixed notion of forcing, $\lambda > 2^{|P|}$ and let $M \prec (H_{\lambda}, \in, <, ..)$, where $H_{\lambda}$ is the set of all sets whose transitive closure has size less than $\lambda$, and $' <'$ is a well-ordering of $H_{\lambda}$. Then a condition $q \in P$ is $(M,P)$-generic if for every maximal antichain $A \in M$, the set $A \cap M$ is predense below $q$.

He moves on to state a small lemma, saying that its proof is a routine verification (that leads us to the assertion that I am not an old hand):

Assume the same situation as in our definition above. Then the following are equivalent:

(i) $q$ is $(M,P)$-generic.

(ii) $q \Vdash \dot{G} \cap M$ is a filter on $P$ generic over $M$

The (ii) $\Rightarrow$ (i) direction is quite easy but I couldn't find a proof for (i) $\Rightarrow$ (ii).

Thank you

1

Equivalent definitions of $(M,P)$-genericity

Hi!

I started to read the chapter 31 in Jechs book about proper forcing. Unfortunately it is written in a rather sketchy way and I do have some issues in proving a lemma about two equivalent definitions of $(M,P)$-genericity. Jech gives the following definition:

Let $(P,<)$ be a fixed notion of forcing, $\lambda > 2^{|P|}$ and let $M \prec (H_{\lambda}, \in, <, ..)$, where $H_{\lambda}$ is the set of all sets whose transitive closure has size less than $\lambda$, and $He moves on to state a small lemma, saying that its proof is a routine verification (that leads us to the assertion that I am not an old hand): Assume the same situation as in our definition above. Then the following are equivalent: (i)$q$is$(M,P)$-generic. (ii)$ q \Vdash \dot{G} \cap M$is a filter on$P$generic over$M$The (ii)$\Rightarrow$(i) direction is quite easy but I couldn't find a proof for (i)$\Rightarrow\$ (ii).

Thank you