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 $\lambda^{\text{cf}(\kappa)}<\kappa$.
Best Answer
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 $\kappa$.)