I'm reading Cummings' chapter in the handbook of set theory, and I have a question regarding the following forcing+assertion:
" Let $κ = cf(κ) ≥ ω_2$. We define a forcing poset $P$ which aims to add a non-reflecting stationary set of cofinality $ω$ ordinals in $κ$, that is to say a stationary $S ⊆ κ ∩ Cof(ω)$ such that $S ∩ α$ is non-stationary for all $α ∈ κ ∩ Cof(> ω)$. $p ∈ P$ if and only if $p$ is a function such that
$1.$ $dom(p) < κ$, $ran(p) ⊆ 2$.
$2.$ If $p(α) = 1$, $cf(α) = ω.$
$3.$ if $β ≤ dom(p)$ and $cf(β) > ω$ then there exists a set $c ⊆ β$ club in $β$ such that $∀α∈c p(α)= 0.$
It is easy to see that $P$ is countably closed, and that it adds the characteristic function of a stationary subset of $κ$."
[ This is Example 6.5 and its on page 21 in the link from his site: http://www.math.cmu.edu/users/jcumming/papers/repaper_finished_june_2008.pdf ]
I dont see why the generic function is a characteristic of a stationary, any help would be appreciated, thanks.
I think this is the same argument as for showing that a ($\kappa$)-Cohen real is a stationary set:
Given any name for a club $\dot C \subseteq \kappa$ and any condition $p$, define a decreasing sequence $\langle p_n \rangle_{n \in \omega}$ below $p$, and a strictly increasing sequence of ordinals $\langle \alpha_n \rangle_{n \in \omega}$ so that $p_n \Vdash \alpha_n \in \dot C$ and $\alpha_{n+1} > \operatorname{dom} p_n$. Then any lower bound $q$ of $\langle p_n \rangle_{n \in \omega}$ forces that $\alpha := \sup \alpha_n \in \dot C$, because $\dot C$ is forced to be club. But note that in fact $q := \bigcup_{n \in \omega} p_n$ is a lower bound of $\langle p_n \rangle$ and $\alpha \notin \operatorname{dom} q$. Also, $\alpha$ has countable cofinality. So we can extend $q$ to $r := q \cup \{ (\alpha,1) \}$. But then $r \Vdash \alpha \in \dot S \cap \dot C$ where $\dot S$ is the name for the generic.