I'm looking at König's Lemma and Aronszajn Trees. I've seen the following link: König's Infinity Lemma and Aronszajn Trees
Long story short, I'm still left with two questions and I was wondering if anyone would either be able to point me in the right direction or explain why. I apologise if they're silly questions, I'm just very curious and can't quite deduce/find the answer myself.
1) Why does a proof in the style of König's Lemma break down for $\omega_{1}$? What is wrong with "Pick a node that has an uncountable number of successors as, if one doesn't exist, then you have a countable union of countable things which must (choice) be countable."? (Just to save time, why doesn't whatever goes wrong here also go wrong in the finite case? I've read it's about cardinal exponentiation, but don't quite understand.)
2) Does the fact that CH implies there exists an $\aleph_{2}$ Aronszajn Tree simply follow from theorem 7.10 in Kanamori's book (about $\kappa$ being regular and $2^{<\kappa}$ = $\kappa$) or is there an explicit construction I might be able to find somewhere? (or is the explicit construction simply the proof with $\aleph_2$ instead of $\kappa$?) My problem here is about CH. As far as I'm aware, $\aleph_{1}$ is regular using choice, so does CH give us the condition that $2^{<\aleph_{1}}$ = $\aleph_{1}$ and hence there is an $\aleph_{2}$ Aronszajn Tree?
Thank you in advance for any help!
For the $\omega_1$ case, the reason is that at limit stages if we collect all the branches, we'll have uncountably many points, so we have to allow only a few branches to go through limit steps.
As for the second question, $\sf CH$ gives us exactly that $\aleph_1^{<\aleph_1}=\aleph_1^{\aleph_0}=\aleph_1$, which is why there is an $\aleph_2$-Aronszajn tree. As for an explicit construction, you can start with these assumptions, and work through Specker's proof to have an explicit one.