Forcing Over CH

Let $\mathbb P = Fn(\omega_2\times\omega,2)$ be the collection of all the finite partial functions from $\omega_2\times \omega$ to $2$. Our strategy is:

• to firstly find a collection of dense sets $D_{\alpha\beta}$, such that a generic filter $G$ can be build upon;
• to secondly prove that any generic filter of $\mathbb P$ preserves cardinals.

Lemma. 1 $D_{\alpha\beta}$ are dense sets, where
$$D_{\alpha\beta} = {p\in\mathbb P\mid \exists n\in\omega(\langle\alpha,n\rangle\in dom(p), \langle\beta,n\rangle\in dom(p),p(\alpha,n)\neq p(\beta,n))}.$$

Thus, for any generic filter $G\subseteq \mathbb P$, $G\cap D_{\alpha\beta}\neq\emptyset$ and thus $\bigcup G$ is a function $\omega_2\times\omega\to 2$.

Given that $\mathbb P$ satisfies c.c.c.$^1$, we have

Lemma. 2 $\mathbb P$ preserves cofinality and thus preserves cardinals.

Proof. Let $\kappa$ be a regular cardinal which witnesses the change of cofinality. Thus, there are $\alpha<\kappa$ and $f:\alpha\to \kappa,f\in V[G]$ which maps $\alpha$ cofinally into $\kappa$. Given that $\mathbb P$ satisfies c.c.c., we can find a $F:\alpha\to P(\kappa)$ in $V$ such that:

• $\forall\xi<\alpha(f(\xi)\in F(\xi))$;
• $\forall\xi<\alpha(|F(\xi)|\leq\omega)^{V2}$.

Thus, let $S = \bigcup_{\xi<\alpha}F(\xi)$, we have that $S$ is unbounded in $\kappa$, and $|S| = |\alpha| = \kappa$, contradicting the regularity of $\kappa$.
$\square$

By the above lemmas, for any generic filter $G\subseteq V$, $\bigcup G$ is a function from $\omega_2\times\omega\to 2$, which results in $\omega_2$ many different $\omega\to 2$ functions. And since $\omega_2$ is preserved in $V[G]$, we may say that $V[G]\vDash \neg CH$.

$^1$ About the c.c.c. property of $\mathbb P$: Suppose $A = {p_\alpha\mid \alpha<\omega_1}$ is an uncountable antichain of $\mathbb P$, then it is safe to assume that ${dom(p_\alpha)}$ is a Delta system. Let $r$ be its root, then we can forwardly assume that all $p_\alpha$ agrees on $r$. Thus all $p_\alpha$ are compatible with each other.

$^2$ See Lemma. 5.5 of Chap. VII of Kunen’s book.