Another Update

Okay, so I’ve actually finished working through the proof that any inaccessible Jónsson cardinal must be Mahlo ($\omega$-Mahlo even), but I haven’t had much time to write stuff up. The main issue is that I’ve also been working my way through Sh413, which contains the result that any $\lambda$ which is an inaccessible Jónsson cardinal must be $\lambda\times\omega$-Mahlo.

What I’ll try to do is write up the proof that any inaccessible Jónsson cardinal must be Mahlo. That’s a bit more work than the thread I was following earlier, but not by much. One thing I will do is take the existence of the desired club guessing sequences for granted. I feel moderately comfortable in doing this because the construction of these sequences is incredibly similar to the construction in the case that $\lambda=\mu^+$ in EiSh819.

From there, I want to try and write up the result of working through the first section of Sh413. There’s a lot of material in that first section which ends up being tertiary to the main result, and I would like to write up a “straight shot” proof of the result. In particular, there is a really cool construction of a club guessing sequence provided that we have a stationary set $S$ which does not reflect outside of itself (Claim 0.14 from Sh413). Actually I might write that part up sooner rather than later.