In a set theory book, I read a proof of
There exists an Aronszajn tree
It makes use of the following lemma.
Let $[\omega_1]^2$ be the collection of all increasing pairs of countable ordinals. Suppose e is a function from $[\omega_1]^2$ into some set $X$. Then, for each $\alpha < \omega_1$ we write $e_\alpha : \alpha \rightarrow X$ for the function with domain $\alpha$, given by $e_\alpha(\beta) = e(\langle \beta,\alpha\rangle)$.
Then there is $e : [\omega_1]^2\rightarrow\omega$ satisfying the following conditions
- For every $\alpha < \omega_1$, $e_\alpha$ is one-to-one.
- For all $\alpha < \beta < \omega_1$, $e_\alpha$ and $e_\beta\upharpoonright\alpha$ disagree in only nitely-many places.
I understand the proof, but later it says, that a similar argument can be used to show that
CH implies the existence of an $\omega_2$-Aronszajn tree.
I was giving a try. I think I should try toprove first a similar version of the lemma by replacing $\omega_1$ by $\omega_2$, $\omega$ by $\omega_1$, and "finitely-many" by "countably many". But I can't reach the conclusion.
Any help is welcome, thanks