Forcing theory / Cardinal preservation

Question

I am going through the forcing theory and there is a proposition/exercise stated on the lectures, but without proof.

I was wondering if someone could give me a help (or hint) how to do it.

Namely, work over a countable ground model $M$. Let $P$ be the forcing Fn$(\aleph_1,P(\omega),\aleph_0)$ defined as usual, i.e.

Fn$(\aleph_1,P(\omega),\aleph_0)=${$p|p:dom(p) \to P(\omega),dom(p)\subset\aleph_1,card^M(dom(p))<\aleph_0$}.

$1)$ Which $M$-cardinals are preserved by forcing with $P$?

$2)$ What is the value of $2^{\aleph_0}$ in $P$-generic extensions of $M$?

Thanks in advance for any help.

Answer 1

Kunen VII.6.10: $Fn(I,J,\lambda)$ has the $(|J|^{<\lambda})^+$-c.c.

Kunen VII.6.9: Assume $P \in M$, $\theta$ is a cardinal of $M$, and ($P$ has the $\theta$-c.c.) in $M$. Then $P$ preserves cofinalities $\ge \theta$. Hence if $\theta$ is regular in $M$, then $P$ preserves cardinals $\ge \theta$.

Putting those together, your $P$ preserves $M$-cardinals $\ge |P(\omega)|^+$ (i.e., cardinals $> |P(\omega)|$), which partially answers question 1. That's as far as I can get...

Answer 2

Your forcing makes $\mathcal{P}(\omega)$ countable. To show this we'll use the following fact: that if $\beta$ is an ordinal and every $\gamma\le \beta$ has countable cofinality, then $\beta$ is countable (since we can't have $\omega_1\le\beta$).

So fix $\lambda$ any limit ordinal $\le 2^{\aleph_0}$, and let $f: \lambda\rightarrow\mathcal{P}(\omega)$ be injective. Let $g$ be the function $\aleph_1\rightarrow\mathcal{P}(\omega)$ gotten from your forcing, and define a partial function $h: \subseteq\omega\rightarrow\lambda$ given by

• $h(n)=\alpha$ if $g(n)=f(\alpha)$, and

• $h(n)$ is undefined otherwise.

By genericity, the range of $h$ is cofinal in $\lambda$; but then $\lambda$ has cofinality $\omega$.

Since every $\lambda<2^{\aleph_0}$ has cofinality $\omega$ in the generic extension, this means that the old $2^{\aleph_0})$ is countable in the generic extension.

In combination with Ted's answer, this completely resolves Question $1$.

Answer 3

If you are still interested, I can answer the second question for you. The continuum in $M [G]$ will have size $(2^{2^\omega})^M$. Here $"\leq"$ follows from the standard nice names argument since the forcing has size continuum. For the $"\geq"$ part, let $f=\bigcup G:\omega_1\rightarrow\mathcal {P}(\omega)^M $ be the function added by this forcing. By a standard density argument, $f $ restricted to $\omega$ is already surjective (which also simplifies Noahs argument that the continuum is collapsed). This implies that the map $g:\mathcal {P}(\mathcal {P}(\omega))^M\rightarrow \mathcal{P}(\omega)^{M [G]} $ which maps a set $A$ to its preimage under $f $ restricted to $\omega$ is injective.