An exercise in Kunen

Question

The following exercise appears in Kunen;

In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, and all cardinals of $M$ above $\lambda$ remains cardinals in $M[G]$.

He gives the hint: If $f=\bigcup{G}$, then $f\upharpoonright\omega$ maps $\omega$ onto $\lambda$.

I'm not really sure how you show that. It is certainly true that $f$ is a function from $\kappa$ onto $\lambda$ but I can't really see why the hint would be true.

Also, how do you do the rest of the problem? In the book, preservation of cardinals is shown via showing the preservation of cofinalities. Here cofinalities are not going to be preserved (If I accept the hint, then ordinals of cofinality $\lambda$ in the ground model will have cofinality $\omega$ in the extension).

Answer 1

For each $\alpha < \lambda$, consider $D_\alpha := \{p \in Fn(\kappa, \lambda) : (\exists n \in \omega)(p(n) = \alpha)\}$. Since elements of $Fn(\kappa, \lambda)$ are finite partial functions, each $D_\alpha$ is dense. Therefore by the genericity of the generic filter, $f \upharpoonright \omega$ is a surjection of $\omega$ and $\lambda$. $\lambda$ is countable in the generic extension.

Note that in $V$, $|Fn(\kappa, \lambda)| = \lambda$. Therefore, $Fn(\kappa, \lambda)$ has the $\lambda^+$-cc. Hence all cardinals above $\lambda$ are preserved by a general result about chain conditions (this result is in Kunen).

Answer 2

For the hint: Isn't it true that for any $α ∈ λ$ and any $f ∈ Fn(κ, λ)$ can be extended so it maps some $n ∈ ω$ to $α$? Then $\{f ∈ Fn(κ, λ): α ∈ f \upharpoonright ω\}$ is dense.

And about the cardinal preservation, it should be true that cardinals $≥ λ$ are preserved and their regularity is also preserved, but you probably need to assume that $λ$ is regular. You should also observe and use that your $Fn(κ, λ)$ is $λ^+$-cc.