# Cardinal inequality question used in Silver's theorem proof

In Jech's Set Theory, Theorem 8.13 states:

If the Singular Cardinals Hypothesis holds for all singular cardinals with cofinality $$\omega$$, then it holds for all singular cardinals.

And Lemma 8.14 states:

Let $$\kappa$$ be a singular cardinal such that $$\text{cf}(\kappa)>\omega$$ and $$\lambda^{\text{cf}(\kappa)}<\kappa$$ for all $$\lambda<\kappa$$.

For a normal sequence $$\big\langle\kappa_\alpha:\alpha<\text{cf}(\kappa)\big\rangle$$ with $$\lim \kappa_\alpha=\kappa$$, if the set $$\{\alpha<\text{cf}(\kappa):\kappa_\alpha^{\text{cf}(\kappa_\alpha)}=\kappa_\alpha^+\}$$ is stationary, then $$\kappa^{\text{cf}(\kappa)}=\kappa^+$$.

In Jech, the proof of the Theorem 8.13 goes as follows: We prove this theorem by induction on the cofinality of $$\kappa$$. Let $$\kappa$$ be a singular cardinal that has an uncountable cofinality. One can show by induction on $$\lambda$$ that $$\lambda<\kappa$$ implies $$\lambda^{\text{cf}(\kappa)}<\kappa$$. Then for a normal sequence

$$\big\langle\kappa_\alpha:\alpha<\text{cf}(\kappa\big\rangle$$

such that $$\lim\kappa_\alpha=\kappa$$, note that the set

$$S=\big\{ \alpha<\text{cf}(\kappa):\text{cf}(\kappa_\alpha) =\omega\text{ and }2^{\aleph_0}<\kappa_\alpha \big\}$$

is a stationary set in $$\text{cf}(\kappa)$$. By our hypothesis, we know $$\alpha\in S$$ implies $$\kappa_\alpha^{\text{cf}(\kappa_\alpha)}=(\kappa_\alpha)^+$$. So by the Lemma 8.14, we have $$\kappa^{\text{cf}(\kappa)}=\kappa^+$$.

I wanted to ask few things about the above proof. First, does "induction on cofinality of $$\kappa$$" mean: assume SCH holds for all singular cardinals with cofinality $$<\text{cf}(\kappa)$$, and we will show that SCH holds for all singular cardinals with cofinality $$=\text{cf}(\kappa)$$?

Secondly, how does one prove that $$\lambda<\kappa$$ implies $$\lambda^{\text{cf}(\kappa)}<\kappa$$ using our induction hypothesis and also induction on $$\lambda$$? When $$\lambda$$ is a successor cardinal, I was fine, but when $$\lambda$$ is a limit cardinal, and moreover, if $$\text{cf}(\lambda)=\text{cf}(\kappa)$$, then I am not sure why $$\lambda^{\text{cf}(\kappa)}<\kappa$$.

• Use $\{$ to produce $\{$ Jan 22, 2023 at 3:48
• Thanks it worked! It didn't work before for some reason. Jan 22, 2023 at 3:51

As far as I can tell, this is just an error in Jech: the proof should be by induction on $$\kappa$$, not by induction on $$\operatorname{cf}(\kappa)$$. This solves your issue with the case $$\operatorname{cf}(\lambda)=\operatorname{cf}(\kappa)$$, since then the induction hypothesis is that SCH holds for all cardinals below $$\kappa$$ so $$\lambda^{\operatorname{cf}(\kappa)}=\lambda^{\operatorname{cf}(\lambda)}=\lambda^++2^{\operatorname{cf}(\lambda)}<\kappa$$.

(For the full details of the proof, you can refer, as Jech does, to Theorem 5.22(ii). That theorem is stated with SCH as a hypothesis, but its proof for fixed $$\kappa$$ only uses SCH for cardinals less than or equal to $$\kappa$$. So, in the proof of Theorem 8.13, you can apply Theorem 5.22(ii) to compute $$\lambda^{\operatorname{cf}(\kappa)}$$ for any $$\lambda<\kappa$$, since the induction hypothesis is that SCH holds for all cardinals below $$\kappa$$.)