If $\kappa$ is a regular cardinal then $\kappa^{<\kappa} = \max\{\kappa, 2^{<\kappa}\}$

Question

I want to show that, assuming AC, that if $\kappa$ is a regular cardinal then $\kappa^{<\kappa} = \max\{\kappa, 2^{<\kappa}\}$ where $\kappa^{<\lambda}$ is defined by $$\kappa^{<\lambda} = \sup\{\kappa^{\nu}\mid\nu \in Card , \nu < \lambda\}.$$
If $\kappa= \aleph_0$ then $\aleph_0^{<\aleph_0} = \sup\{\aleph_0^n\mid n \in \omega\} =\aleph_0^{\aleph_0} = 2^{\aleph_0} = \max\{\kappa, 2^{\kappa}\} = \max\{\kappa,2^{<\kappa}\}$.

If $\kappa$ is a successor cardinal, then $\kappa = \aleph_{S(\alpha)} = \aleph_{\alpha + 1}$ and, using Hausdorff's formula, we have $$\aleph_{\alpha+1}^{<\aleph_{\alpha +1}} = \sup\{\aleph_{\alpha+1}^{\aleph_{\beta}} \mid \beta < \alpha + 1\} = \sup\{\max\{\aleph_{\alpha+1},\aleph_{\alpha}^{\aleph_{\beta}}\} \mid\beta < \alpha + 1\} = \max\{ \aleph_{\alpha+1}, \sup\{{\aleph_{\alpha}^{\aleph_{\beta}}\mid\beta < \alpha + 1\}}\} = \max\{\aleph_{\alpha+1},\aleph_{\alpha}^{\aleph_{\alpha}}\}$$ and, since $\aleph_{\alpha}^{\aleph_{\alpha}} = 2^{\aleph_{\alpha}} = 2^{<\kappa}$ this case is proved.

I cannot prove the result for $\kappa$ limit. If $\kappa$ is limit and regular cardinal then it is weakly inaccessible, i.e. $\kappa = \aleph_\kappa$. I'm asking an hint for this part and if the rest of the proof is correct.

Answer 1

Your proof about $\aleph_0$ is definitely not correct. Note that $\aleph_0^n=\aleph_0$ for all $n>0$. Therefore $\sup\{\aleph_0^n\mid n\in\omega\}$ is literally $\sup\{1,\aleph_0\}=\max\{1,\aleph_0\}$. And I will let you figure that part out on your own.

Now, for the successor case, note that since this function is increasing, $\aleph_{\alpha+1}^{<\aleph_{\alpha+1}}=\aleph_{\alpha+1}^{\aleph_\alpha}$. This is equal to $\aleph_{\alpha+1}\cdot 2^{\aleph_\alpha}$, and incidentally $2^{\aleph_\alpha}=2^{<\aleph_{\alpha+1}}$, so the result follows.

But actually this proof also translates to the inaccessible case. If $\kappa$ is regular, limit or otherwise, and $\lambda<\kappa$, then every function from $\lambda$ to $\kappa$ is bounded. So $\kappa^\lambda$ is exactly $\sup\{\alpha^\lambda\mid\alpha<\kappa\}$.

Now, if $\kappa$ is a successor say $\lambda^+$, then this just $\kappa\cdot\lambda^\lambda=\kappa\cdot2^\lambda=\kappa\cdot2^{<\kappa}$; but if $\kappa$ is a limit cardinal, then we get $\kappa^\lambda$ to be $\sup\{\mu^\lambda\mid\mu<\kappa\text{ regular}\}$ which by the inductive hypothesis is eventually (read: for $\mu>\lambda$) equals to either $\mu$ or $2^\lambda$; since there are $\kappa$ many options for $\lambda$, we get exactly $\sup\{2^\lambda\mid\lambda<\kappa\}$ which is at least $\kappa$, but also exactly $2^{<\kappa}$ by definition.