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

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 $\kappa^{++}.$

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).

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  • $\begingroup$ This is quite recent, so I don't think there are any tangible material written yet. Yair told me that he figured out a proof just about when Magidor told him that they cracked it; so maybe he could shed some light on the matter. $\endgroup$ – Asaf Karagila Mar 1 '15 at 22:55
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    $\begingroup$ So, Magidor is coaxing Foreman back into forcing and large cardinals? $\endgroup$ – Monroe Eskew Mar 2 '15 at 2:24
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    $\begingroup$ I think you might be able to watch Dima Sinapova lecturing on the tree property at: newton.ac.uk/event/hifw01 $\endgroup$ – Avshalom Aug 30 '15 at 11:08
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The following theorem from Sinapova's paper ''THE TREE PROPERTY AT THE FIRST AND DOUBLE SUCCESSORS OF A SINGULAR'' 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 $\kappa_0^{++}$.

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  • $\begingroup$ @MohammadGolshani you're welcome! $\endgroup$ – Rahman. M Aug 30 '15 at 3:55

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