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.



Tree property at weak inaccessibles
In his paper [Boolean extensions which efface the Mahlo property](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/boolean-extensions-which-efface-the-mahlo-property1/34045FCF82EA845DDB2F3A1E6D393211) 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?


