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Forcing Axioms and the CH by Aspero/Larson/Moore

On page 11 of this paper I struggle with the beginning of the proof of Lemma 3.5.


Coding:

For $r \in 2^\omega$ and $A \in H(\aleph_1)$ let us say $f$ codes $A$ if $(TC(A), \in, A) \cong (\omega, R_1, R_2)$, where $R_1 \subseteq \omega^2$ and $R_2 \subseteq \omega$ are defined by

$(i,j) \in R_1 \Leftrightarrow r(2^{i+1}(2j+1)) = 1$,

$i \in R_2 \Leftrightarrow r(2i+1) = 1$.

Let $f$ a finite-to-one function from a set of ordinals with order-type $\omega$ into $2^{< \omega}$, we say that $f$ codes $A \in H(\aleph_1)$ if, for some cofinite subset $X \subset dom(f)$, $\bigcup f[X]$ is a single infinite lenghth sequence which codes $A$ in the sense above.


$\psi_2$:

$\psi_2$ is the assertion that for every ladder system $\bar{C}$, every triple $\omega_1 < \alpha < \beta < \gamma < \omega_2$ of ordinals, and every $\omega_1$-club $\mathcal{N}$ in $[\gamma]^{\aleph_0}$, there is a function $f \colon \omega_1 \to 2^{< \omega}$ such that for every limit $\delta < \omega_1$, $f \upharpoonright C_\delta$ codes the transitive collapse of a structure $(N, \in, \omega_1, \alpha, \beta; X_i \colon i < \omega)$, where $\lbrace X_i \rbrace_{i < \omega}$ is an increasing sequence in $\mathcal{N}$ of height greater than $\delta$ and $N = \bigcup_{i < \omega}X_i$.


Forcing $Q$:

The ground model may satisfy $2^{\aleph_0} = \aleph_1$ and $2^{\aleph_1} = \aleph_2$. Let $\vec{C}$ a ladder system on $\omega_1$, $\omega_1 < \alpha < \beta < \gamma < \omega_2$ and $\mathcal{N}$ an $\omega_1$-club in $[\gamma]^{\aleph_0}$. We define $Q = Q_{\vec{C}, \alpha, \beta, \mathcal{N}}$ to be the collection of all $q$ such that $dom(q) = \eta$ for some countable limit ordinal $\eta$, $q$ maps into $2^{<\omega}$, and $q$ satisfies the conclusion of $\psi_2$ for $\delta \leqslant \eta$. It is $\vert Q \vert = 2^{\aleph_0} = \aleph_1$.


I want to know if this partial order $Q$ satisfies the hypothesis of the following lemma. Especially if for $\xi < \omega_1$, the set $\lbrace q \in Q \colon \vert q \vert \geqslant \xi \rbrace$ is dense in $Q$.

Lemma: Let $X \subseteq H(\aleph_1)$ and $Q \subseteq X^{< \omega_1}$ (ordered by extension), such that

  1. $Q$ is closed under initial segments;
  2. for every $\alpha < \omega_1$, $\lbrace q \in Q \colon \vert q \vert \geqslant \alpha \rbrace$ is dense;
  3. if $q \in Q$ with $\vert q \vert = \alpha, p \colon \alpha \to X$ and $\lbrace \xi < \alpha \colon q(\xi) \neq p(\xi) \rbrace$ is finite, then $p \in Q$.
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    $\begingroup$ Questions here should really be as self-contained as possible. You should edit the question to include the pertinent information, including the basic definition of the forcing, and the statement/hypotheses of Lemma 3.4. Links should only be used as reference, not to contain nearly all of the content of your question. $\endgroup$ – user642796 Aug 14 '15 at 11:02
  • $\begingroup$ I thought it might be too long first but I edited my question now, thank you. $\endgroup$ – os23 Aug 16 '15 at 21:43

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