Timeline for Veblen function with uncountable ordinals & beyond
Current License: CC BY-SA 4.0
9 events
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| May 31, 2020 at 13:49 | comment | added | Simply Beautiful Art | An inaccessible cardinal $I$ is regular and a fixed-point of $x\mapsto\omega_x$, which ensures that $F:I^\omega\to I$ is definable on all inputs (i.e. we never go beyond $I$). | |
| May 31, 2020 at 13:41 | comment | added | SSequence | of halt-values (of suitable programs) as an analogue of $\omega_1$ (for "container functions"). But I haven't really thought about it carefully enough. Regarding collapsing functions, I don't know anything about them (except that they are based on closure) ..... but anyway, that's besides the point. Anyway, I hope this comment of mine wasn't too confusing. | |
| May 31, 2020 at 13:41 | comment | added | SSequence | @SimplyBeautifulArt "and the analog of $\omega_1$ for the modified Veblen function would be an inaccessible cardinal or something of similar nature" ...... Interesting observation. Yeah I wasn't sure (since I don't know anything about bigger cardinals etc.) about what would be the large-enough "container value" (analogous to $\omega_1$) when we are considering "storage function" in an extended manner such as in OP. That's why I left this point. Though I am reasonably certain that for the cases such as in OP we should be able to take some kind of supremum (continued) | |
| May 31, 2020 at 13:04 | comment | added | Simply Beautiful Art | The use of $F$ and $\omega_1$ as you describe is more or less equivalent to Chris Bird's ordinal collapsing function, and the analog of $\omega_1$ for the modified Veblen function would be an inaccessible cardinal or something of similar nature. | |
| May 31, 2020 at 7:14 | history | edited | SSequence | CC BY-SA 4.0 |
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| May 31, 2020 at 5:49 | comment | added | SSequence | I do not know. A lot of them are quite technical. The usage of infinite programs here is quite basic. I am merely suggesting they can preserve the details of operations (in the same natural number) of using $x \mapsto \omega^x$ to go to $\Gamma_0$ and using $x \mapsto \omega_x$ to go to "analgoue of $\Gamma_0$" which you mention in the question. | |
| May 31, 2020 at 5:39 | history | edited | SSequence | CC BY-SA 4.0 |
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| May 31, 2020 at 4:24 | comment | added | user820789 | Nice approach! Do you have any good resources for infinite programs? | |
| May 31, 2020 at 4:13 | history | answered | SSequence | CC BY-SA 4.0 |