Forcing CH but not adding $\omega_1$-sequences

Question

Given a ctm $M$, is there a forcing which does not add $\omega_1$-sequences, and forces CH in $M[G]$?

Answer 1

Unless I'm misunderstanding the question, the answer is no.

I assume that a forcing extension $M\subseteq M[G]$ "adds $\omega_1$-sequences" if there is some map $f:\omega_1^M\rightarrow M$ which is in $M[G]$ but not $M$. If so, then we can argue as follows (with $M\models \neg$ CH, since the other possibility is uninteresting):

• If $\omega_1^M$ is countable in $M[G]$, then we've added a new real, so a fortiori a new $\omega_1$-sequence.

• So suppose $\omega_1^M=\omega_1^{M[G]}$. If $M[G]\models$ CH, there is in $M[G]$ some surjection $f$ from $\omega_1^{M[G]}$ to $\mathbb{R}^M$; but since $\omega_1^M=\omega_1^{M[G]}$, the function $f$ is an $\omega_1$-sequence in the sense of $M$, and since $M\models\neg$ CH it must be a new $\omega_1$-sequence.

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EDIT: As per Joel's comment below, note that forcing plays no role here: any time I can extend a model of set theory to one in which CH holds without adding $\omega_1$-sequences (in the sense of the original model), the original model must also satisfy CH. The relevant notion of "extension" here is end extension: $M\subseteq_{end}N$ iff $M$ is a substructure of $N$ and for each $x\in M$ we have $\{y\in M: y\in^Mx\}=\{y\in N: y\in^Nx\}$ (that is, no objects in $M$ "get new elements" when we pass to $N$).

Exercise: show that forcing extensions are in fact end extensions!