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

## Critical Cardinals

Yup. I posted a new paper on arXiv. And if you're one of my three regular readers, you know that I am not going to talk about the paper itself (I leave that to the paper), but rather about the process leading to it. If you don't care, that's fine, the paper is on arXiv and you can check the Papers section of the site to see if it's been published or whatnot.

So, this one has been on the back burner for a while. And it actually started as two separate projects that merged and separated and merged again. Continue reading...

## The transitive multiverse

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins' multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, $\{(M,E)\}$ would be an entire multiverse. That's quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit... off putting. Continue reading...

## No uniform ultrafilters

Earlier this morning I received an email question from Yair Hayut. Is it consistent without the axiom of choice, of course, that there are free ultrafilters on the natural numbers but none on the real numbers?

Well, of course that the answer is negative. If $\cal U$ is a free ultrafilter on $\omega$ then $\{X\subseteq\mathcal P(\omega)\mid X\cap\omega\in\cal U\}$ is a free ultrafilter on $\mathcal P(\omega)$. But that doesn't mean that the question should be trivialized. What Yair asked was actually slightly subtler than that: is it consistent that there are free ultrafilters on $\omega$, but no uniform ultrafilters on the real numbers? Continue reading...