# 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

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.