While I was dealing with Aronszajn trees I found the following exercise from Kunen's old book.
If $\kappa=\lambda^+$ and $\lambda$ is a regular uncountable cardinal and $2^{<\lambda}=\lambda$ holds then there exists a $\kappa$-Aronszajn tree.
The author says that to prove the result it's enough to apply the same argument used to see the existence of Aronszajn trees (i.e, by means of a coherent sequence $\{s_\alpha\}_{\alpha<\omega_1}$) but now assuming that $ran(s_\alpha)$ is contained in a non stationary set. My questions are the following:
- Regarding the case $\kappa=\omega_2$, the construction of the functions $s_\alpha$ with $\alpha$ of countable cofinality can be the following. Let $\{\alpha_n\}_n$ such that $\alpha_n\rightarrow \alpha$ and $\{s_{\alpha_n}\}_n$ a sequence such that $\forall\,n\in\omega\,\forall \beta\in\alpha\,(\beta<\alpha_n\longrightarrow\,s_{\alpha_n}\upharpoonright \beta=s_\beta$). Now define a sequence of $t_n$ (each one in the $\alpha_n-$level of the tree $T=\{s\in \omega_1^{<\omega_1}:\, s\,\text{is one to one}\}$) such that
- $t_0=s_{\alpha_0}$
- $t_n=^*s_{\alpha_n}$ where $=^*$ means equal but countable many places.
- $t_{n+1}\upharpoonright\alpha_n=t_n$
Finally take $t=\bigcup_n t_n$ and $s_\alpha=t$. It's easy to see that $\forall\beta<\alpha,\, s_\alpha\upharpoonright \beta=^* s_\beta$. On the other hand $ran(s_\alpha)$ is contained in a non estationary since $\bigcup_{n\in\omega}ran(s_{\alpha_n})$ is non stationary and $\bigcup_{n\in\omega} (ran(t)\setminus ran(s_{\alpha_n}))$ is countable and thus also non stationary.
But, what about those function whose $\alpha$ are of uncountable cofinality as $s_{\omega_1}$? Could someone show me an explicit construction for that kind of functions?
- In the general case $\kappa=\lambda^+$, why is nedeed regularity of $\lambda$?
Any help or commentary will be helpful to me.