Reading guidance request to understand this lemma in set theory Sorry if this question does not meet the requirements of a proper question in MSE; in this case I would appreciate any guidance on how to ask appropriately or where I could get an answer. I am reading through this article by Shelah and I cannot understand the last lemma - lemma 13. Here is the statement and the proof,



Where $\Gamma = \{0,1\}^{\omega}$ is considered as a subspace of the Baire space $\omega^\omega$, and $\mathscr{E}$ is an analytic equivalence relation on $\Gamma$ is meant to be that $\mathscr{E}$ is an analytic subset of $ \Gamma \times \Gamma$.
I just know some elementary set theory (at the level of cardinals and ordinals) and some elementary logic and model theory (compactness theorem), and do not know any forcing or descriptive set theory, which I think are relevant subjects. More exactly I don't know the meaning of the words/symbols elementary submodel of $( H((2^{\aleph_0} )^+), \mathscr{E})$, Cohen generic over $N$, absoluteness criteria and $N[A_1, A_2]$. Regarding the amount that I don't know about this lemma, it seems that I should (would better) start reading some things more comprehensively than reading wikipedia pages. I would appreciate helps and reading suggestions to gain a good understanding of this lemma.
 A: I shall try to explain what the specific parts you mentioned mean, whilst trying to avoid set theory as much as I can, but I recommend reading up on these notions from text books if you don't feel comfortable with them. For most things you can look at Jech's Set Theory book, but for forcing, I recommend Kunen's An introduction to independence proofs book.


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*"an elementary submodel of $(H((2^{\aleph_0})^+), E)$": this is just a model theoretical notion. First just in case you are not familiar with $H((2^{\aleph_0})^+)$, note that for any regular cardinal $\kappa$, we define $$H(\kappa) = \{x: Card(trcl(\{x\})) < \kappa\}.$$ So it is the set of all sets such their transitive closure has size less than $\kappa$. You can show that this set exists and in your case $H((2^{\aleph_0})^+)$ includes all the relevant information about the reals, the Baire space, etc. Now we say some $N\subset H((2^{\aleph_0})^+)$ is an elementary submodel of $(H((2^{\aleph_0})^+), E)$, just in case that for any formula $\varphi(x_1, \dots, x_n)$ which may mention $E$ as a predicate and for any set of parameters $a_1, \dots, a_n$ from $N$, we have $$N\models\varphi(a_1, \dots, a_n)\leftrightarrow H((2^{\aleph_0})^+)\models\varphi(a_1, \dots, a_n).$$ So this $N$ sees the same truths as $(H((2^{\aleph_0})^+), E)$. Also you can see that $E\in H((2^{\aleph_0})^+)$. And so by using elementarity you can also show that $E\in N$.


*"Cohen generic over $N$": this is just forcing. But in the lemma, you can find an equivalent formulation. $x$ is Cohen generic over $N$, if and only if for every meager(countable union of nowhere dense sets) set of $\Gamma$(note that by elementarity and the fact that $\Gamma$ is definable, $\Gamma \in N$) which lives in $N$, like $A$, we have that $x\not\in A$. Note that although $\Gamma\in N$, it is not the case that $\Gamma\subset N$, since $N$ is countable. And also any Cohen generic real $x$ cannot be an element of $N$(can you see why?), so it is not unreasonable for such $x$ to exist. In fact via a simple forcing argument you can see that since $N$ is countable, many such $x$ in fact do exist!


*"$N[A_1, A_2]$": this is just the smallest model of set theory that contains $N$ as a subset and $A_1, A_2$ as elements.(Well, this is not entirely true, but you can think of it this way. The thing is that in general the structure $N[A_1, A_2]$ is defined to be the forcing extension of $N$ via two consecutive extensions with the Cohen forcing and it may not satisfy ZFC as $N$ does not satisfy full ZFC. But at this point, since you are not familiar with forcing, you can think of it  as a model of ZFC.)


*"absoluteness": this is a general notion in set theory. So suppose we have two models $M\subset N$ of set theory. We say a formula $\varphi(x_1, \dots, x_n)$ is absolute between $M$ and $N$, if for any set of parameters $a_1, \dots, a_n$ from $M$, $$M\models\varphi(a_1, \dots, a_n)\leftrightarrow N\models\varphi(a_1, \dots, a_n).$$ And in your case, note that since you are dealing with an analytic set, it is definable via a boldface $\Sigma^1_1$ formula, and by a result from descriptive set theory we know that $\Sigma^1_1$ formulas are absolute between models of set theory so if for example if some things are $E$-equivalent in a larger model, they will also be $E$-equivalent in the smaller models.
Hope this helps.
