Timeline for Cardinality of infinite towers of Alephs - can tower be more than countable?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2021 at 12:47 | vote | accept | tzimie | ||
| Mar 1, 2021 at 18:07 | answer | added | Noah Schweber | timeline score: 4 | |
| Mar 1, 2021 at 17:46 | comment | added | Noah Schweber | @tzimie "Wiki claims that $\epsilon_1$ can be defined as infinite tower of $\epsilon_0$" Not quite - we also need to throw in some "$+1$"s to make it work. Specifically, $$\epsilon_1=\sup\{\epsilon_0+1, \omega^{\epsilon_0+1}, \omega^{\omega^{\epsilon_0+1}}, ...\}.$$ That "$+1$" might look minor but it makes a huge difference. (See e.g. here.) | |
| Mar 1, 2021 at 14:01 | comment | added | tzimie | @AlecRhea yes, so there is a huge gap between $T(\omega)$ and first inaccesible. | |
| Mar 1, 2021 at 13:59 | comment | added | tzimie | @Wojowu Wiki claims that $\epsilon_1$ can be defined as infinite tower of $\epsilon_0$, but $\epsilon_0$ is an infinite tower of $\omega$, so repeating $\omega$ exponentiation $\omega$ times we get $\epsilon_0$, but repeating it more we get $\epsilon_1$ | |
| Feb 28, 2021 at 0:15 | comment | added | Alec Rhea | An inaccessible can’t have the same cofinality as any ordinal smaller than it, and these things you’re defining have smaller cofinalities by definition. I don’t think you can reach any really big ordinals this way. | |
| Feb 27, 2021 at 16:15 | review | Close votes | |||
| Mar 3, 2021 at 21:11 | |||||
| Feb 27, 2021 at 15:59 | comment | added | Wojowu | RE first comment, well of course $T(\omega)^+$ is a larger cardinal, but to claim it is a larger tower you would need to define what counts as a tower, which gets us back to square one. RE second comment, the Wikipedia article you link at no point mentions any towers, let alone ones of transfinite length. | |
| Feb 27, 2021 at 15:25 | comment | added | tzimie | @Wojowu for the tower of epsilons, I found it in Wiki en.wikipedia.org/wiki/Epsilon_numbers_(mathematics) | |
| Feb 27, 2021 at 15:24 | comment | added | tzimie | @Wojowu Thank you, you're right, of course, it is not possible for any tower to reach first inaccessible. But $T(\omega)$ can't be the highest tower as well, because $T(\omega)^{+}$ is bigger, so there should be many cardinalities between $T(\omega)$ and the first inaccessible - probably, these cardinalities can be coded by the parameter of T greater than $\omega$ | |
| Feb 27, 2021 at 13:27 | comment | added | Wojowu | I'm not aware of any general definitions of "tower of length [whatever]" which would let $\epsilon_0^{\epsilon_0^{\epsilon_0}}$, do you know of a definition which achieves that? I don't think the problem is with limitation of ZFC to only having finite formulas or being first-order, it's about it being unclear what such a tower should mean. To address your last question: I'm not sure what $\theta_0$ is, but $T(\omega)$, the limit of $T(n),n<\omega$, is way smaller than the first inaccessible. | |
| Feb 27, 2021 at 13:08 | history | edited | gmvh |
Added top-level tag
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| Feb 27, 2021 at 12:49 | comment | added | David Roberts♦ | Similar things have been asked elsewhere, eg math.stackexchange.com/questions/2689417/… and math.stackexchange.com/questions/1747465/… . See also en.m.wikipedia.org/wiki/… | |
| Feb 27, 2021 at 11:50 | comment | added | tzimie | Another thought, isn't the first inaccessible cardinal a limit of all towers, so $\theta_0 = T(\omega)$ ? In such case $T(\omega+1) = \theta_{\aleph_0}$ ? | |
| Feb 27, 2021 at 11:45 | comment | added | tzimie | @Wokowu I dont think about any exact definition, I am just trying to understand why there is such limitation. Any ideas are welcome. So what you are saying is that adding one more aleph doesn't change a height of a tower. $1+\omega = \omega$. But with ordinals we can go beyond $\epsilon_0$ by building higher tower of $\omega$ : $\epsilon_0^{\epsilon_0^{\epsilon_0}}$. Can we do the same with $\aleph_0$ ? Anyway, what if the tower is the size of continuum? | |
| Feb 27, 2021 at 11:35 | comment | added | SSequence | @Wojowu I suppose one choice could be $T(\omega+1)=\aleph_{T(\omega)+1}$? Of course, I do not know whether this is what the OP had in mind. | |
| Feb 27, 2021 at 11:08 | comment | added | Wojowu | It's not clear how you would extent $T$ nontrivially to infinite ordinals - if you let $T(\omega)$ be the limit of $T(n),n<\omega$, and try to define $T(\omega+1)$ as $\aleph_{T(\omega)}$, then you will find $T(\omega+1)=T(\omega)$, and by this process you will find $T(\alpha)=T(\omega)$ for all $\alpha\geq\omega$. Do you have some different definition in mind? | |
| Feb 27, 2021 at 10:51 | review | First posts | |||
| Feb 27, 2021 at 13:47 | |||||
| Feb 27, 2021 at 10:46 | history | asked | tzimie | CC BY-SA 4.0 |