## Henselianity in the language of rings, with Franziska Jahnke

I would like to write about a new paper ([AJ15]) which Franziska Jahnke and I have written and put on the arXiv. It’s called Henselianity in the language of rings. In it we investigate the relationship between the following four properties of a field $K$:

(h) $K$ is henselian, i.e. $K$ admits a nontrivial henselian valuation,

(eh) $K$ is elementarily henselian, i.e. every $L\equiv K$ is henselian,

(def) $K$ admits a definable nontrivial henselian valuation, and

($\emptyset$-def) $K$ admits a $\emptyset$-definable nontrivial henselian valuation.

## Puzzle corner: Solution to Sylvy’s puzzle #4

Cross-reference to puzzle page on my website

In this post I want to discuss the solution to Puzzle #4. Let’s recall the puzzle: it was to find an uncountable chain in $(\mathcal{P}(\mathbb{N}),\subseteq)$.