# Does $\kappa \lt \lambda$ implie $cf\kappa \lt cf\lambda$ for singular cardinals?

In T. Jech's set theory, the part which he tries to prove a part of silver's theorem on the singular cardinal hypothesis, namely,
"If the singular cardinal hypothesis holds for all singular cardinals of cofinality $\omega$, then it holds for all singular cardinals."
Is confusing me a little bit. He assumes that $\kappa$ is a singular cardinal of uncountable cofinality and he uses induction on the cofinality of $\kappa$ to prove the statement. The part which i am struggling to understand is where he says that for each $\lambda \lt \kappa$ we have $\lambda^{cf\kappa} \lt \kappa$. I know that assuming the singular cardinal hypothesis $\lambda^{cf\kappa}$ is either $2^{cf\kappa}$ or $\lambda$ or $\lambda^{+}$ based on the continuum and cofinality functions. So if induction on the cofinality of $\kappa$ is the same as induction on singular cardinals then i can assume that the singular cardinal hypothesis holds bellow $\kappa$ and i'm fine. So the question is:
Does $\kappa \lt \lambda$ implie $cf\kappa \lt cf\lambda$ for singular cardinals?

No. Recall that if $\delta$ is a limit ordinal, then $\operatorname{cf}(\aleph_\delta)=\operatorname{cf}(\delta)$.
Now take $\kappa=\aleph_{\omega_1}$ and $\lambda=\aleph_{\omega_1+\omega}$.
As for the proof of Silver's theorem, note that if $\lambda<\kappa$, by induction on $\lambda$, and using both your assumption and Hausdorff's formula to obtain that $\lambda^{\operatorname{cf}(\kappa)}<\kappa$.
• How can i use my induction hypothesis for limit $\lambda$? Commented May 5, 2018 at 7:04
• I'm sorry, I don't have the time to give you an answer off hand with full details. I think that it would be best if you post a separate question . As for the horizontal line, --- three dashes in a separate line would create that. I also recommend paragraph breaks (which are two consecutive line breaks), as they make reading much easier, and this question of yours is kind of hard on the eyes. Commented May 5, 2018 at 9:53