## 116c- Lecture 7

We presented the proof that “$k$-trichotomy” implies choice. The following is still open:

Question. (${\sf ZF}$) Assume that $X$ is non-well-orderable. Is there a countably infinite family of pairwise size-incomparable sets?

We mentioned a few (familiar) statements that fail in the absence of choice, like the existence of bases for any vector space, Tychonoff’s theorem, or the “surjective” version of the Schröder-Bernstein theorem.

We defined addition, multiplication and exponentiation of cardinals, and verified that addition and multiplication are trivial. We stated the continuum hypothesis ${\sf CH}$, and the generalized continuum hypothesis ${\sf GCH}$.

We want to prove (in subsequent lectures) a few non-trivial results about the behavior of exponentiation. In order to do this, we need the key notion of cofinality. We proved a few basic facts about cofinality and defined regular cardinals.