Timeline for The Tall Tale of Terminating Transfinite Towers
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 18, 2018 at 9:01 | comment | added | Morteza Azad | @MohammadGolshani Interesting thesis which partially serves as an expository paper on Kaplan-Shelah's result! Kheili Mamnun, Mohammad! :-) | |
| Jul 18, 2018 at 5:39 | vote | accept | Morteza Azad | ||
| Jul 18, 2018 at 3:00 | comment | added | Mohammad Golshani | You may also look at Automorphism towers and definability in generalized Baire spaces where details about the Kaplan-shelah's result and related theorems are proved. | |
| Jul 17, 2018 at 15:50 | comment | added | Morteza Azad | (+1) While it is rather hard to believe that Simon's seemingly strong result is provable merely using $ZF$ but let's trust Itay, Saharon, and the anonymous referees of their paper, and take this fact for granted. I am inclined to push the acceptance button for this answer (to the question 1) right now, but maybe it is better to wait a little bit to see whether anything shows up for the question 2. Anyway, thanks for the useful reference, Asaf! | |
| Jul 17, 2018 at 14:23 | comment | added | Asaf Karagila♦ | @Burak: Ah, okay. | |
| Jul 17, 2018 at 14:16 | comment | added | Burak | @AsafKaragila: Yes, the paper proves Simon's result without AC, so, this together with Joel's paper, which apparently not uses AC, answers the question. When I read Joel's comment, I mistakenly thought that Joel's 1998 proof of the full result did not use AC (since this paper of Shelah was written in 2009.) That's why I wanted to double check whether there was some way to avoid Simon's proof back then in Joel's 1998 paper. | |
| Jul 17, 2018 at 14:11 | comment | added | Joel David Hamkins | My theorem is that for any group $G$, the tower eventually has a centerless group, and this does not use AC. I am interpreting Asaf's answer here to say that Simon's theorem can be proved for centerless groups without AC. In this case, the full termination theorem can be proved in ZF. | |
| Jul 17, 2018 at 14:11 | comment | added | Asaf Karagila♦ | @Burak: Have you read my answer and at least the abstract of the linked paper? | |
| Jul 17, 2018 at 14:08 | comment | added | Burak | @JoelDavidHamkins: Joel, doesn't your proof reduce the general case to the centerless case? If so, Simon's proof, as far as I know, uses AC. | |
| Jul 17, 2018 at 13:58 | comment | added | Joel David Hamkins | I think my proof does not use AC at all, and so from any group $G$ you get eventually (by the replacement axiom) a centerless group. So this seems to complete the proof in ZF. | |
| Jul 17, 2018 at 13:52 | history | answered | Asaf Karagila♦ | CC BY-SA 4.0 |