5

I've seen this statement in multiple posts (e.g. here and here), but I can't seem to understand it. I can see why

$$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0},$$

by noting that every $\alpha<\omega_2$, so $|\alpha|\leq\aleph_1$, meaning

$$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\max_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0}.$$

So how does $\aleph_2$ enter the equation?

Community
  • 1
Dan Saattrup Nielsen
  • 988

2 Answers2

7

The sum has $\aleph_2$ terms, each of which is at most $\aleph_1^{\aleph_0}$, so it’s bounded above by $\aleph_2\cdot\aleph_1^{\aleph_0}$. On the other hand, it’s clearly at least $\aleph_2$ and at least $\aleph_1^{\aleph_0}$, so its bounded below by their maximum, which is simply their product.

Brian M. Scott
  • 616,228
5

We can prove something quite more general. Given a regular cardinal $\kappa$, noticing that each function $\lambda\rightarrow \kappa$ for $\lambda<\kappa$ ($\lambda$ cardinal) must have bounded range in $\kappa$, we can write $$\kappa^{\lambda}=\bigcup_{\alpha <\kappa} \alpha^{\lambda},$$ where $\alpha$ ranges over ordinals. Therefore we get easily $$\kappa^{\lambda}=\sum_{\alpha< \kappa}\vert \alpha\vert^{\lambda}.$$ Using this fact, since each successor cardinal is regular, we can prove (for all $\alpha,\beta$ ordinals) the Hausdorff's Formula: $$\aleph_{\alpha +1}^{\aleph_{\beta}}=\aleph_{\alpha +1}\cdot\aleph_{\alpha}^{\aleph_{\beta}}.$$ (The question is a particular case for $\alpha =1$ and $\beta =0$).

Marco Vergura
  • 4,436
  • Thanks for the answer. Why is it important that $\kappa$ is regular? Does it have something to do with the boundedness of the range of functions $f\in{^\lambda\kappa}$? – Dan Saattrup Nielsen May 20 '13 at 11:30
  • @Leidem Yes, it does: if $\kappa$ is regular, its cofinality, $cf(k)$, is $\kappa$ itself and therefore you can't have a cofinal map from a cardinal $\lambda< \kappa$ to $\kappa$ (otherwise $cf(k)\leq \lambda$), i.e. given any cardinal $\lambda< \kappa$ there are no functions $\lambda\rightarrow \kappa$ with unbounded range. Boundness of functions is the key point to be able to write down $\kappa$ as that union (it is exactly what that means). – Marco Vergura May 20 '13 at 12:06
  • Ah yeah, that makes sense, thank you! On a small note, I think you meant $\alpha<\kappa$ in the union. – Dan Saattrup Nielsen May 20 '13 at 13:35
  • Yes, for sure, thanks! I'll edit! – Marco Vergura May 20 '13 at 13:40