# Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.

1. $\mathbb{P}$ preserves stationary subsets of $\omega_1$
2. $\mathbb{P}$ fails to have the $\omega$-covering property. So in particular, $\mathbb{P}$ cannot be $\sigma$-distributive, or proper on any stationary set of countable models.
3. Whenever $\kappa$ has uncountable cofinality, then this remains true in $V^{\mathbb{P}}$.

Item (3) rules out Namba forcing,Prikry forcing, and the standard stationary tower forcings with critical point at least $\omega_2$ (the latter preserve stationary subsets of $\omega_1$ if the height of the tower is a Woodin cardinal, but change lots of regular cardinals to cof $\omega$).

Item (2) rules out several forms of antichain sealing forcings (all variants of sealing forcings I know of are of the form "proper followed by shooting a club", which has the countable covering property).

• This seems somewhat close to recent work of Yair Hayut and yours truly. Although there's still quite a gap, if you require that $\Bbb P$ do not add new subsets of $\omega_1$, and that it is a "tiny bit homogeneous" (in an odd, technical sense), then it not change cofinalities at all. – Asaf Karagila Jul 15 '15 at 19:55

## 1 Answer

Let $(\kappa_n: n<\omega)$ be an increasing sequence of measurable cardinals with limit $\kappa,$ and for each $n$ let $U_n$ be a normal measure on $\kappa_n.$ Let $\mathbb{P}$ be the Prikry type forcing notion which adds one element Prikry to each $\kappa_n:$

$p\in \mathbb{P}$ iff $p=(s, A),$ where

1)$s$ is a finite sequence,

2) $\forall i< |s|, s(i) < \kappa_i,$

3) $A=(A_n: |s| \leq n < \omega),$ and $A_n \in U_n$,

$p=(s,A)\leq q=(t, B)$ iff:

1) $s$ end extends $t$,

2) $\forall |t| \leq i < |s|, s(i)\in B_i,$

3) $\forall |s| \leq n < \omega, A_n \subseteq B_n.$

The forcing adds an $\omega$-sequence to $\kappa,$ and can not be covered by any ground model set of size $\omega$ (in fact of size $< \kappa$), it adds no bounded subsets to $\kappa,$ and does not change cofinalities.

So the forcing is as requested.

• Do you know if the consistency strength of this violation of $\omega$-covering is exactly $\omega$ many measurable cardinals? – Yair Hayut Jul 16 '15 at 15:18
• Unfortunately I have no idea. – Mohammad Golshani Jul 17 '15 at 4:54
• It can be done with just a single measurable cardinal: just take a quotient of Prikry forcing by another Prikry forcing on co-infinitely many coordinates. – Bill Chen Jul 18 '15 at 22:39