Timeline for Veblen function with uncountable ordinals & beyond
Current License: CC BY-SA 4.0
16 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jun 6, 2020 at 10:52 | history | bounty ended | user820789 | ||
| May 29, 2020 at 23:43 | comment | added | Simply Beautiful Art | @meowzz We can chat in here if you'd like. | |
| May 29, 2020 at 23:24 | comment | added | user820789 | Chat would be great. | |
| May 29, 2020 at 23:09 | comment | added | Simply Beautiful Art | @meowzz With what? And I think we're kind of going a bit far off topic. If you want to, we could enter a chat room. | |
| May 29, 2020 at 22:52 | comment | added | user820789 | How would you annotate the large Veblen ordinal? | |
| May 29, 2020 at 22:43 | comment | added | Simply Beautiful Art | @meowzz The Veblen function on transfinitely arguments works the same way as I have already described. With your fixes though, if I am understanding them correctly, $\phi_0''(0)$ would simply be $\phi(\omega@\omega)$. It is not clear what you mean to have after that, but everything is probably not past the Veblen function on $\omega+k$ arguments for maybe $k=2$ or $k=3$. | |
| May 29, 2020 at 22:29 | comment | added | user820789 | I just want to note that I already understood what you put about the multivariable Veblen function. I just am not as sure about the notation of the version with transfinitely many variables. After looking a bit more I found something that put it as $\phi(1@\omega)$ which seems logical to me. I may modify this post soon to integrate that notation. My idea was that $\phi_0''(0)$ would be the supremum of the transfinitely many variables version. Also, while $\phi_\alpha'(\beta)$ might have similar strength (more, less or equal) to the normal function modified w/ $\omega_1$, what about $\Xi$? | |
| May 28, 2020 at 12:53 | comment | added | Simply Beautiful Art | @FedorPakhomov That is likely true, but I don't remember off the top of my head if Rathjen had used $\Phi$ there. | |
| May 28, 2020 at 11:45 | comment | added | Fedor Pakhomov | @SimplyBeautifulArt Rathjen's notation system for $\mathsf{KPM}$ definitely covers this cardinals. But as I pointed out in my comment to this question, this cardinals would be already present if we collapse below the first inaccessible. | |
| May 27, 2020 at 23:05 | comment | added | Simply Beautiful Art | I would recommend steering away from fundamental sequences. These ordinals are not even countable, so even if they aren't regular, they may not have countable cofinality. Personally, I find it a lot simpler to work with the concept of closure: "what operations can apply to smaller ordinals that will not result in ordinals larger than $\phi_{1,0}'(0)$?" As I mentioned, I've seen it used in Rathjen's papers. I can try to pull this up later. Per the aside, it was made up on a whim. | |
| May 27, 2020 at 22:50 | history | edited | Simply Beautiful Art | CC BY-SA 4.0 |
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| May 27, 2020 at 22:50 | comment | added | user820789 | Thanks for the feedback! I had a feeling I was not annotating $\phi_{1,0}'(0)$ correctly. I updated the equation, however am still unsure if this is the proper fundamental sequence. Wrt "the usual Veblen function modified w/ $\phi(\alpha)=\omega_\alpha$" - do you know of any literature that does this? I've only seen the idea mentioned a few times online & haven't actually seen anyone work through it. (Aside: did you create the $\omega_\star$ notation here or does it come from somewhere?) | |
| May 27, 2020 at 22:49 | history | edited | Simply Beautiful Art | CC BY-SA 4.0 |
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| May 27, 2020 at 21:25 | history | edited | Simply Beautiful Art | CC BY-SA 4.0 |
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| May 27, 2020 at 21:20 | history | answered | Simply Beautiful Art | CC BY-SA 4.0 |