# Set Theory exercise.

I find myself unable to solve question 24.1 of T. Jech's Set Theory:

If $\beta<\omega_1$ and if $2^{\aleph_{\alpha}}\leq\aleph_{\alpha+\beta}$ for a stationary set of $\alpha$'s, then $2^{\aleph_{\omega_1}}\leq\aleph_{\omega_1+\beta}$.

[By induction on $\beta$: If $\varphi(\alpha)\leq\beta$ on a stationary set, then $||\varphi||\leq\beta$.]

I am unable to prove the hint (in brackets). Any hints for the hint?

-
While MathOverflow is not the place for asking help with exercises in general, I do think that questions from chapter 24 of Jech can make the exception, and vote against closing. (Although this question would definitely be welcomed on math.stackexchange.com just as well) –  Asaf Karagila May 19 '13 at 20:27
Presentation is the key, a different title would have helped avoid confusion. Most exercises in Jech are definitely on topic here though there is a useful trick which is to dig through the papers in the historical notes until a solution is found. For Chapter 24, this is not too helpful since reading Shelah's book on pcf theory is not for the light hearted. (Fortunately, I think this one is from the older Galvin-Hajnal paper.) –  François G. Dorais May 19 '13 at 22:04
I can recommend on "Introduction To Cardinal Arithmetic" by Holtz, et. al for the Galvin-Hajnal stuff. I learned the basics from there one time and it was very readable and clear. I also agree with Francois that presentation is the key, the standards for book exercise questions on MO are definitely higher than those of math.SE! –  Asaf Karagila May 19 '13 at 22:08

Recall the definitions (24.1) and the consequence of the proof of Lemma 24.2, which states that $\|\varphi\|=\sup\lbrace\|\psi\|+1\mid\psi<\varphi\rbrace$.
Then we have that $\|\varphi\|=0$ if and only if $\varphi(\alpha)=0$ on a stationary set. Proceed by induction; suppose this claim is true for all $\delta<\beta$ and let $\varphi$ be such that $\varphi(\alpha)\leq\beta$. Clearly if $\varphi(\alpha)\leq\delta<\beta$ then we are done, so suppose that $\varphi(\alpha)=\beta$ for a stationary set.
So if $\psi<\varphi$ we have that on a club set, $\psi(\alpha)<\beta$, and therefore has rank of at most $\delta<\beta$. So the rank of $\varphi$ is at most $\beta$.