# Downward Löwenheim-Skolem Theorems and Choice Principles

I have posted a new note on the Papers page.

It’s a short little proof that the classic downward Löwenheim-Skolem theorem is equivalent to $\DC$, and that for a well-ordered $\kappa$, the downward Löwenheim-Skolem asserting the existence of models of cardinality $\leq\kappa$ is in fact equivalent to the conjunction of $\DC$ and $\AC_\kappa$.

The proof is quite straightforward and not very long.

Interestingly enough, despite not appearing on the “usual” choice dictionaries, this was known for quite some time. It appeared in a book by George Boolos, and it was independently found by Christian Espindola (not too long ago as well!). You can find his versions on his homepage.

Any comments on the note, or suggestions for improvements and extensions are more than welcome here on this page.

Downward Löwenheim-Skolem Theorems and Choice Principles.

# …And we’re back!

Okay, I took the time to make some changes to my homepage.

Clearly, the theme is different now. I also changed the content of the Papers page. I removed the abstracts (for some reason I thought this is going to be a cool thing to have, but with time it grew to annoy me greatly). I will definitely post a few things there in the coming time, some notes and eventually some nice papers — I hope!

I also put back some of the blog posts. About half didn’t make the cut, but that’s fine. Nothing really important in them anyway.

Now back to work… or something.

(P.S. I’m kinda of annoyed by the fact that the text width is so… small. There’s way too much dead space on my monitor right now.)