Forcing GCH with iteration of $\mathsf{Add}(\kappa,1)$

Question

For a successor cardinal $\kappa^+$, forcing with $\mathsf{Add}(\kappa^+,1)$ forces $2^\kappa=\kappa^+$ in the extension. Can one use an iteration of this kind of forcing to for the GCH? If so, I'd like to know if there is a write-up of this proof somewhere. In particular, what kind of support should it have? Is there a characterization of what cardinals are preserved or collapsed?

Answer 1

It is well-known that iterating $\mathsf{Add}(\kappa^+,1)$ with Easton support iteration (or reverse Easton iteration) forces $\mathsf{GCH}$. Here Easton support iteration means the following:

• If $\alpha$ is a singular limit, then take the inverse limit, and
• If $\alpha$ is a regular limit, then take the direct limit.

You may find some part of the proof from Sy Friedman's Fine Structure and Class Forcing. I believe there is a full proof somewhere, but I have no idea where it is. Here is the proof I sketched:

We will clarify the definition of the iteration we will use. Let $\langle \mathbb{P}_\alpha \mid \alpha\in\mathsf{Ord}\rangle$ be the iteration of $\langle\dot{\mathbb{Q}}_\alpha\mid\alpha\in\mathsf{Ord}\rangle$, where

• If $\alpha$ is a cardinal in $V^{\mathbb{P}_\alpha}$, then $\dot{\mathbb{Q}}_{\alpha}=\mathsf{Add(\alpha^+,1)}$, and
• If $\alpha$ is not a cardinal in $V^{\mathbb{P}_\alpha}$, then $\dot{\mathbb{Q}}_{\alpha}$ is a trivial forcing.

Let $\mathbb{P}$ be the Easton support iteration of all $\langle \mathbb{P}_\alpha \mid \alpha\in\mathsf{Ord}\rangle$. Then $\mathbb{P}$ preserves $\mathsf{ZFC}$. (Using the Easton support is necessary to prove it.) Then we have the following property:

Theorem. Let $\gamma$ be an ordinal such that $V^{\mathbb{P}_\gamma}$ thinks $\gamma$ is regular. Then $V^{\mathbb{P}_\gamma}$ thinks the tail forcing $\dot{\mathbb{P}}_{\gamma,\infty}$ is $(\gamma^+)$-closed. (Note that $\mathbb{P}\cong \mathbb{P}_\gamma*\dot{\mathbb{P}}_{\gamma,\infty}$).

Now we prove $\mathbb{P}$ forces $\mathsf{GCH}$ as follows:

Proof. Assume that $\mathbb{P}_\beta$ forces $\mathsf{GCH}$ below $\beta$ for all $\beta<\alpha$. Furthremore, assume that $\mathbb{P}_\alpha$ forces $\alpha$ is a cardinal. (We do not need to consider when $\mathbb{P}_\alpha$ forces $\alpha$ is not a cardinal.)

• If $\mathbb{P}_\alpha$ forces $\alpha=\beta^+$ for some cardinal $\beta<\alpha$, then $\mathbb{P}_\beta$ forces $\beta$ is a cardinal. Furthermore, $\mathbb{P}_{\beta+1}$ forces $2^\beta=\beta^+$. Hence $\mathbb{P}_\alpha=\mathbb{P}_{\beta+1}$ forces $\mathsf{GCH}$ below $\alpha$.
• Now assume that $\mathbb{P}_\alpha$ forces $\alpha$ is a limit ordinal. Fix $\beta<\alpha$ which is a cardinal in $V^{\mathbb{P}_\alpha}$, and take $\gamma$ such that $V^{\mathbb{P}_\alpha}\models \gamma=\beta^+<\alpha$. Then $\mathbb{P}_\gamma$ forces $\gamma$ is regular (since $\mathbb{P}_\alpha$ does and $\gamma<\alpha$.) Since $\mathbb{P}_\alpha\cong \mathbb{P}_\gamma*\dot{\mathbb{P}}_{\alpha,\gamma}$, $V^{\mathbb{P}_\gamma}\models 2^\beta=\beta^+<\alpha$ and ($V^{\mathbb{P}_\gamma}$ thinks) $\dot{\mathbb{P}}_{\alpha,\gamma}$ is $\gamma^+$-closed, we have $V^{\mathbb{P}_\alpha}\models 2^\beta=\beta^+<\alpha$.

We can see that inaccessible cardinals are preserved. But some cardinals may collapse; for example, if the ground model satisfies $2^{\aleph_0}=\aleph_2$, then $\mathsf{Add}(\aleph_1^V,1)$ forces $2^{\aleph_0}=\aleph_1$ while preserving all cardinals below $\aleph_1^V$ and above $\aleph_3^V$, so it collapses $\aleph_2^V$.