I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:



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> **Theorem.** It is consistent, relative to the existence of large cardinals, that there is a singular strong limit cardinal $\kappa,$ such that the tree property hold at both $\kappa^+$ and $\kappa^{++}.$
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Here are some questions related to the above theorem:


1) Does anyone know the main idea of the proof? 


2) Is the $\kappa$ in the theorem large, or it can be also small, like $\aleph\_\omega?$


3) What kind of large cardinal(s) are used in the proof?


4) Are there any notes or slides presenting the proof (as far as I know the paper itself is not written yet, as I asked Magidor for the paper, and did not receive anything).



Tree property and singular strong limit cardinals
The following theorem from Sinapova's paper ''[THE TREE PROPERTY AT THE FIRST AND DOUBLE
SUCCESSORS OF A SINGULAR](http://homepages.math.uic.edu/~sinapova/Single%20and%20double%20successors.pdf)'' answers your main question.


**Theorem.** Suppose that $\langle\kappa\_{n}:n<\omega\rangle$ is an increasing sequence of super-compact cardinals and that $\lambda$ is a weakly compact cardinal with $\lambda>\sup\{\kappa\_n:n<\omega\}$. Then there is a generic extension in which $\kappa\_0$ is strong limit singular with $cf(\kappa\_0)=\omega$ and $\lambda=\kappa\_0^{++}$, and the tree property holds at both $\kappa\_0^{+}$ and $\kappa\_0^{++}$.


