Timeline for The Tall Tale of Terminating Transfinite Towers
Current License: CC BY-SA 4.0
14 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 18, 2018 at 5:39 | vote | accept | Morteza Azad | ||
| Jul 17, 2018 at 16:34 | comment | added | Joel David Hamkins | This is in part what Gunter Fuchs and I had done. In my argument with Simon, we had to force to create the group, but with Gunter, we proved that there is such a malleable group assuming only $\Diamond$. We never got it purely in ZFC. | |
| Jul 17, 2018 at 16:09 | comment | added | Morteza Azad | @JoelDavidHamkins A side remark on your joint paper with Simon: It seems your result is slightly weaker than saying "The terminating number is forcing-sensitive". In fact, you have proved that it is possible (i.e. consistent) for the terminating number to be forcing sensitive in a suitable generic extension. Right? If so, was there any progress in the direction of obtaining a more direct result? I mean changing the terminating number from an arbitrary ground model. What about indestructibility results in the form of forcing terminating number to be unchangeable under certain forcing notions? | |
| Jul 17, 2018 at 15:31 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jul 17, 2018 at 15:08 | comment | added | Morteza Azad | @JoelDavidHamkins Yes, I am well aware of that paper (and also your related work with Fuchs) which reveals a high degree of sensitivity of the terminating number to forcing and set-theoretic assumptions. It is actually a very malleable mathematical object as you described. However, it seems at least in some special cases one may find explicit expressions describing $\tau (G\times H)$. e.g. For $G=H$, a cyclic group of prime order $p$, we have $Aut(G^2)\cong GL_{2}(\mathbb{F}_p)$. So $\tau(G^2)=\tau(GL_{2}(\mathbb{F}_p))+1$ and this holds in every model. | |
| Jul 17, 2018 at 14:32 | comment | added | Joel David Hamkins | For example, in the product case, sure, the product of the automorphism groups is sitting inside the automorphisms of the product, but there could be more, since the two factors could interact, and this might totally affect the nature of the next group in the tower, and therefore could cause huge changes in the height of the new tower. | |
| Jul 17, 2018 at 14:30 | comment | added | Joel David Hamkins | I don't expect any positive results of that nature, in light of the kind of thing happening in my joint paper with Simon: jdh.hamkins.org/changingheights. The phenomenon is that when you make a change to the automorphism group, the tower can suddenly grow much much taller (or shorter). We used this to control the height of a tower of a fixed group by forcing, by adding generic outer automorphisms to the group. | |
| Jul 17, 2018 at 14:26 | comment | added | Morteza Azad | Any thoughts on the second question? | |
| Jul 17, 2018 at 13:54 | comment | added | Joel David Hamkins | Yes, of course, and both forms of closure are equivalent to replacement/collection over separation etc. My proof, however, used explicitly a certain function on the ordinals, the function mapping $\alpha$ to $\beta$, when all the group elements at stage $\alpha$ that will eventually die, die before $\beta$. Meanwhile, it seems that one will want at least power set to define the tower, since otherwise you can't seem to know that the automorphism group even exists as a set. | |
| Jul 17, 2018 at 13:52 | answer | added | Asaf Karagila♦ | timeline score: 12 | |
| Jul 17, 2018 at 13:41 | comment | added | Asaf Karagila♦ | @Joel: Every function from $V$ to $V$ has a closure point, assuming Replacement. | |
| Jul 17, 2018 at 13:33 | comment | added | Joel David Hamkins | My proof that every group leads eventually to a centerless group uses the replacement axiom, since I use that every function from the ordinals to the ordinals has a closure point. I haven't ever thought much about whether the arguments use AC. Hmmm. | |
| Jul 17, 2018 at 13:22 | history | asked | Morteza Azad | CC BY-SA 4.0 |