I am doing some models of ST exercises and I just got stuck with the following one:
Let (($\mathbb P_\alpha,\leq_\alpha,1 _\alpha$)| $\alpha \leq \omega$) be the finite support iteration of the sequence (($\overset . {\mathbb Q}_n,\overset . {\leq_n}$)| $n\in \omega$).
Let $\kappa\geq2$ be a cardinal in the ground model $M$ and suppose furthermore that for each $n\in \omega$, the following holds:
$1_n \Vdash_{\mathbb P_n}^M "\overset . {\mathbb Q}_n$ has an antichain of size $\kappa$."
How can I show that then every $\mathbb P_\omega$-generic extension $M[G]$ contains a surjective function $f: \omega \to \kappa$ which is not in $M.$
For every $n$, let $\dot A_n$ be an antichain of size $\kappa$, and let $\dot a_{n,\alpha}$ denote its $\alpha$th element.
Now, consider the function defined by $f(n)=\alpha$ given by $G\cap\dot A_n=\{\dot a_{n,\alpha}^G\}$ (or $0$ if undefined). As the iteration is a finite support iteration, you can show that this function is not in the ground model $M$ and it is surjective.