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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2$\begingroup$ Note that for any two cardinals $\mu \geq\kappa$, $\mu^\kappa = cf([\mu]^\kappa, \subseteq) \cdot 2^\kappa$ so the first statement implies the second. $\endgroup$– Yair HayutApr 26, 2015 at 15:24
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1 Answer
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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.
answered Apr 27, 2015 at 3:39
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$\begingroup$ Could you send me a reference to the statement "SSH iff $u(\kappa,\lambda)\le \lambda^+$ for all $\kappa\le \lambda$, with $\lambda$ regular" (I did not know it) $\endgroup$ Apr 27, 2015 at 13:23
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$\begingroup$ @LajosSoukup It is stated in the paper mentioned above. Please see the introduction of the paper. $\endgroup$ Apr 28, 2015 at 5:00
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$\begingroup$ Thank you. Concerning the origin of this result I asked Assaf, and he wrote me that this statement also appeared in Sh400B (Shelah: Cardinal Arithmetics for sceptics) $\endgroup$ Apr 28, 2015 at 14:06