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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?

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    $\begingroup$ 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) $\endgroup$
    – Asaf Karagila ♦
    Commented May 19, 2013 at 20:27
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    $\begingroup$ 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.) $\endgroup$
    – François G. Dorais
    Commented May 19, 2013 at 22:04
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    $\begingroup$ 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! $\endgroup$
    – Asaf Karagila ♦
    Commented May 19, 2013 at 22:08

1 Answer 1

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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 $\beta$.

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    $\begingroup$ I would like to point out the use of Fodor's Lemma in the last line of the answer to the OP, since it took me a second to realise... $\endgroup$
    – user3462
    Commented May 19, 2013 at 20:40
  • $\begingroup$ Thank you for your answer, the hint is clear to me now. I'll attempt to tackle the exercise itself again tomorrow. You'll find any questions (which I will surely have) on math.stackexchange.com. And thanks for the comment, it took me more than a second to see the use of Fodor's Lemma there. $\endgroup$
    – User84559
    Commented May 19, 2013 at 21:42
  • $\begingroup$ Tristan, you're welcome. Tanmay, thanks for the remark. $\endgroup$
    – Asaf Karagila ♦
    Commented May 19, 2013 at 21:54

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