# Tree property at weak inaccessibles

Suppose $$\kappa$$ is a weakly inaccessible cardinal with the tree property. What can we say about the height of $$\kappa$$? Is it a weakly-hyper-Mahlo of some sort? Does it enjoy some kind of indescribability property? Of course it is weakly compact in $$L$$, but I am interested in what height properties we can say it has in a universe where $$\kappa$$ is not strongly inaccessible.

• By the work of Boos, the least weakly Mahlo cardinal can have the tree property. Dec 21, 2020 at 17:26
• I think it is open if the least weakly inaccessible can have the tree property. Dec 21, 2020 at 17:28
• What is the height of $\kappa$? Dec 21, 2020 at 18:18
• @MohammadGolshani Great! Can you give a reference? Dec 21, 2020 at 22:21
• @AsafKaragila I just mean some description of “how many” cardinals are below $\kappa$, like not only $\kappa$-many but stationary many regulars, and moreover a stationary many with that property, blah blah. Dec 21, 2020 at 22:22

## 1 Answer

In his paper Boolean extensions which efface the Mahlo property William Boos proves the following consistency result:

Theorem. Assume GCH holds and $$\kappa$$ is weakly compact. Then there exists a cardinal preserving generic extension of the universe in which $$\kappa$$ is the least weakly Mahlo cardinal and the tree property holds at $$\kappa$$.

The proof of the theorem is very similar to that of Mitchell, the main difference is that at Mahlo cardinals below $$\kappa$$ he adds a club of singular cardinals into that cardinal.

The following is asked in the paper and is still open:

Question. Is it consistent that the tree property hold at the least weakly inaccessible cardinal?

• In my last visit of Vienna, Yair and I discussed this question and Yair had some ideas, but we could not find any other time to discuss it again. He may give some more information about the question. Dec 22, 2020 at 3:47