Asaf Karagila
I don't have much choice...

## Preserving Properness

I just posted another problem in the problems page. The prize, by the way, is a bottle of port wine, or equivalent. And I truly hope to make good on that prize.

In another problem there, coming from a work with David Asperó, we asked if an $\omega_2$-closed forcing must preserve the property of being proper. Yasou Yoshinobu provided us with a negative answer based on Shelah's "Proper and Improper Forcing" XVII Observation 2.12 (p.826). Take $\kappa$ to be uncountable, by forcing with $\Add(\omega,1)\ast\Col(\omega_1,2^\kappa)$ and appealing to the gap lemma, $(2^{<\kappa})$ is a tree with only $\aleph_1$ branches. It can therefore be specialized by a ccc forcing in that model. The iteration of these three forcing (Cohen real, collapse, specialize) is clearly proper. But now by forcing with $\Add(\kappa,1)$ we must in fact violate the properness of this forcing, which was defined in the ground model, since the new branch is also generic for the tree and will therefore collapse $\omega_1$. Continue reading...

## Open Problems!

I've decided to have a list of open problems on my site. I am no Erdős, nor Hilbert, nor Knuth.

But I want my own problems page, and it's my site. So to celebreate the new website, I created just that. For the first couple of problems, I've chosen to focus on the axiom of choice. And I don't think that I have much choice, but to keep that interest running. But I can promise that this is not the only type of problems that I will add there. Continue reading...