Is it true that the following two statements are equiconsistent?


(1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$


(2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular cardinal $\mu$ 


Thanks.



Are the failure of SCH and "$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular" equiconsistent?
As it is stated by Yair Hayut, in the comment above, statement $(1)$ implies statement $(2)$. Let's show that statement $(2)$ does not imply statement $(1)$. 


Let $u(\kappa, \lambda)=cf(P\_\kappa(\lambda), \subseteq),$ so that $cf([\mu]^{cf(\mu)}, \subseteq)=u(cf(\mu)^+, \mu).$


As it is stated in [Large cardinals and covering numbers](http://journals.impan.pl/cgi-bin/doi?fm205-1-3), one can obtain a model of $u(\omega\_1, \omega\_\omega)>\omega\_{\omega}^+$$+SCH$ by adding $\aleph\_{\omega+1}$ many Cohen reals to a model in which $2^{\aleph\_\omega}=\aleph\_{\omega+2}$ and $GCH$ holds everywhere else.


Of course this does not imply that statements $(1)$ and $(2)$ have different consistency strength.


On the other hand Shelah's Strong Hypothesis (SSH) is equivalent to the statement $u(\kappa, \lambda)\leq \lambda^+$ for all cardinals $\kappa\leq \lambda,$ with $\kappa$ regular. However it is not known whether the failure of SSH is equiconsistent
with that of SCH.


