Cardinal arithmetic in inner models

Question

I know that regular cardinals get computed weirdly in inner models of determinacy (this comes up in Jackson's analysis of the projective ordinals $\delta^1_n$); this is a question about a specific case of this.

Suppose $V$ contains a proper class of Woodins. Can we still have $\omega_2^{L(\mathbb{R})}=\omega_2$? If so, how high up can we have $\omega_n^{L(\mathbb{R})}=\omega_n$?

The second question admits an easy answer: no for $n>2$. $\omega_n^{L(\Bbb R)}$ is singular, so it certainly cannot be $\omega_n$. — Asaf Karagila, Dec 10 '16 at 20:20
Also, it is of course possible that $\omega_2^{L(\Bbb R)}<\omega_2$, even without the requirement on Woodin cardinals: simply note that the collapse of $\omega_2$ to $\omega_1$ does not add reals, so it does not change $L(\Bbb R)$. — Asaf Karagila, Dec 10 '16 at 20:21
@AsafKaragila Yes, that one I knew - it's always possible for $L(\mathbb{R})$ to be bad at arithmetic. — Noah Schweber, Dec 10 '16 at 20:22
Okay. Now it's getting late, and I want to grade this entire pile tonight. So I will turn to my other desk, and ignore all further things for tonight. — Asaf Karagila, Dec 10 '16 at 20:40

Asaf Karagila · Answer 1 · 2021-10-26T18:15:27.040

8

The second question admits an easy negative answer: in models of $\sf AD$ every $\omega_n$ is singular for finite $n>2$.

Kleinberg claimed in the following paper that this is due to Martin, but without reference. He also proved it as Corollary 2.2.

E. M. Kleinberg, ${\rm AD}\vdash $ “the $\aleph _{n}$ are Jonsson cardinals and $\aleph _{\omega }$ is a Rowbottom cardinal”, Ann. Math. Logic 12 (1977), no. 3, 229--248. MR 469769, Zbl 0378.02032.

edited Oct 26 '21 at 18:15

answered Dec 10 '16 at 20:39

Asaf Karagila

405,794

I believe you can also find this result in the book Infinitary Combinatorics and the Axiom of Determinateness, by Eugene M. Kleinberg, Lecture Notes in Mathematics #612, Springer, 2006 (originally published in 1977). – Mitchell Spector Dec 11 '16 at 01:37
Mitchell, I believe that you're correct. Both these are given as references by your mathematical-brother, Arthur Apter in one of his papers. I went with the paper, as it's easier to get those online (legally, anyway). – Asaf Karagila Dec 11 '16 at 03:36
Indeed -- that reference is fine. I just thought I would add the other as well. – Mitchell Spector Dec 11 '16 at 03:38
After all these years, is it still exciting to see people refer to your advisor's work, or even to your work? – Asaf Karagila Dec 11 '16 at 08:55
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Yes, of course :) . It's a little nostalgic too; that book was published while I was still in graduate school (and, as you say, Gene Kleinberg was my advisor). – Mitchell Spector Dec 11 '16 at 09:17

Cardinal arithmetic in inner models

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