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.
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Sign up to join this communityIs 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.
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, 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.