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I read something that seems to suggest the following is true:

If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, \tau_\xi) : \xi < \omega_1\}$ with $p_\alpha \in \mathbb{P}$ and $\tau_\xi$ a $\mathbb{P}$-name such that for all $p \in \mathbb{P}$ and names $\tau$, if $p \Vdash \tau \in 2^\omega$, then there exists some $\xi < \omega_1$ with $p_\xi \leq p$ and $p_\xi \Vdash \tau = \tau_\xi$.

Is statement indeed true? If so, can someone provide a reference or brief sketch of the proof.

I believe the above should imply that if the ground model satisfy $\mathsf{CH}$, then $\mathsf{CH}$ remains true in a generic extension by a $\aleph_1$ size proper forcing.

Thanks.

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  • $\begingroup$ Could you perhaps give some details as to what suggested this to you? $\endgroup$ – tci Jul 31 '14 at 3:00
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This was answered positively by Komjath here under the weaker assumption that $P$ preserves $\omega_1$.

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  • $\begingroup$ You mean that $|\Bbb P|=\aleph_1$ and that it preserves $\omega_1$, right? $\endgroup$ – Asaf Karagila Jul 31 '14 at 20:17
  • $\begingroup$ Yes. Otherwise this is false. $\endgroup$ – hot_queen Jul 31 '14 at 20:34
  • $\begingroup$ Of course. I just wanted to make sure which assumption was weakened (just the properness, not all of them). Side effects of being tired and whatnot. $\endgroup$ – Asaf Karagila Jul 31 '14 at 20:35

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