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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?

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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 $\lambda^{\operatorname{cf}(\kappa)}<\kappa$.

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  • $\begingroup$ Oh, you're right, so how can i go about proving that statement then? $\endgroup$ – Shervin Sorouri May 5 '18 at 6:53
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    $\begingroup$ I added something, and I hope it will help. $\endgroup$ – Asaf Karagila May 5 '18 at 7:01
  • $\begingroup$ How can i use my induction hypothesis for limit $\lambda$? $\endgroup$ – Shervin Sorouri May 5 '18 at 7:04
  • $\begingroup$ (A side question: how do you put that horizontal line in your answer?) $\endgroup$ – Shervin Sorouri May 5 '18 at 7:15
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    $\begingroup$ 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. $\endgroup$ – Asaf Karagila May 5 '18 at 9:53

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