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I have been looking at a proof of a technical forcing lemma, and I have a couple of questions.

Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, $P$ is a notion of forcing.

Lemma: The following conditions are equivalent:
1. $q \in P$ is $N$-generic, and
2. $\forall \dot{\alpha} \in N, ( N \models$ "$\dot{\alpha}$ is a $P$-name for an ordinal" $\rightarrow q \Vdash \dot{\alpha} \in N)$

To prove $2 \rightarrow 1$, let $A \in N$, $N \models $"A is a maximal antichain." In $N$, we enumerate $A = \{ a_{\xi} : \xi < \kappa \}$. Then, we define $\dot{\alpha} = \{ ( \xi, a_\xi ) : \xi < \kappa \}$.

The argument then claims that $\dot{\alpha}$ is a $P$-name for an ordinal. But, isn't it actually a $P$-name for a set containing a single ordinal? So how could we apply 2?

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    I have to say that there is not enough set up. Is $P$ a notion of forcing in $N$? Is it a notion of forcing in $V$ as well? Since $N$ is countable $P\cap N$ is countable what is the relationship between $P\cap N$ and $N$?2012-04-20
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    @user27974 If $\kappa$ is uncountable then $A\not\subset N$.2012-04-20
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    @azarel: I suspect that $N$ thinks that $\kappa$ is uncountable but it really just a countable ordinal (well, at least if $N$ is transitive...)2012-04-20
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    Where is the lemma from?2012-04-22
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    @AsafKaragila I have added the complete setup now, as well as the statement of the Lemma. This is from the book Baroszynski and Judah, Set Theory of the Real Line.2012-04-22
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    Are you going to give an exact citation or should I read the entire book?2012-04-22
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    page 23, under Proper Forcing.. Thanks..2012-04-23
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    @AsafKaragila I'm pinging you to point you to Kuhndog's last comment (which didn't ping you since he didn't add "@")2012-06-08
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    @MattN.: I am pinging you because I wish to thank you for reminding me about this question. Kuhndog, I tried to convince a friend that knows the answer to come and write it up. He did not do that, but he told me a vague (and frankly, unhelpful) sketch of a possible proof. I think that his sketch was not a good answer, but I also do not remember the details to write it up.2012-06-08

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