I'm struggling with this exercise from Jech; Suppose we have a forcing notion $P$ = the set of all functions from a bounded subset of $\kappa$ into $\{0,1\}$ with $\kappa$ singular (ordered by inclusion). I want to show that $P$ collapes $\kappa$ to cf$(\kappa)$.
I don't really see how to approach this problem. Suppose we have a generic $G$ and let $f = \bigcup G$. We are essentially adding a subset of $\kappa$ in the extension since $f$ is a function from $\kappa\to\{0,1\}$. I don't really see how adding a single subset would collapse anything. The standard Levy collapse approach is rather different. Any hints?
Let's look at a concrete example: $\kappa=\omega_\omega$. We'll break the generic filter $G$ into $\omega$-many "chunks," each coding an ordinal $<(\omega_\omega)^V$; every ordinal $<(\omega_\omega)^V$ will have infinitely many opportunities to be so coded, hence by genericity will be one of the $\omega$-many ordinals coded into $G$, and this means that $(\omega_\omega)^V$ will be countable in $V[G]$.
First, let $S=f^{-1}(\{1\})$. (Note that $S, f$, and $G$ are all "absolutely definable" from each other.) We divide $S$ as follows:
$C_0=S\cap \omega$.
$C_1=S\cap (\omega_1^V\setminus\omega)$.
$C_2=S\cap (\omega_2^V\setminus\omega_1^V)$.
...
If you prefer, you can think of $C_i$ as $f\upharpoonright (\omega_i^V\setminus\omega_{i-1}^V)$ with the convention that $C_0=f\upharpoonright\omega$.
Now I'm going to have each $C_i$ "code" an ordinal $<\omega_i^V$. For each $i$, fix (in $V$) an injection $h_i$ from $\omega_i^V$ to $(2^{[\omega_{i-1},\omega_i)})^V$ (with the convention that $\omega_{-1}=0$), and let $\beta_i=h_i^{-1}(C_i)$ if $C_i\in ran(h_i)$ and $\beta_i=0$ otherwise.
Can you argue that - in $V[G]$ - the map $i\mapsto\beta_i$ is a surjection from $\omega$ to $(\omega_\omega)^V$? Note that each $\alpha<(\omega_\omega)^V$ is in the domain of unboundedly many (indeed, cofinitely many) $h_i$s ...