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Given an ordinal $\alpha$, define $\alpha !$ as it follows: $$ \alpha! := \begin{cases} 0! = 1 \\ (\alpha + 1)! = \alpha ! \cdot (\alpha + 1) \\ \lambda! = \left(\sup_{\gamma < \lambda} \gamma !\right) \cdot \lambda &\text{$\lambda$ limit} \end{cases} $$

I was trying to evaluate $\varepsilon_0 !$, where $\varepsilon_0$ is the smallest $\varepsilon$-number (the least fixed point of the function $x \mapsto \omega^x$). I can apply the definition as it follows:

$$ \varepsilon_0 ! = \left(\sup_{\alpha < \varepsilon_0} \alpha ! \right) \cdot \varepsilon_0 $$

now all that remains to be done is to evaluate $\sup\{\alpha ! : \alpha < \varepsilon_0\}$. I claim this to be $\varepsilon_0$, because it's obvious that $\varepsilon_0 = \sup\{\alpha : \alpha < \varepsilon_0\}\leq \sup\{\alpha ! : \alpha < \varepsilon_0\}$, since the map $\alpha \mapsto \alpha !$ is stractly increasing (if I'm not mistaken it's a simple induction). Now I'd like to prove the opposite inequality, which means I'd like to prove that for every $\alpha !$, $\alpha < \varepsilon_0$, there exists an ordinal $\beta < \varepsilon_0$ such that $\beta \geq \alpha!$, in order to do this, I was trying to prove that $\alpha < \varepsilon_0 \implies \alpha ! < \varepsilon_0$, and, since $\varepsilon_0$ is a limit ordinal i can use $\beta = \alpha ! + 1 < \varepsilon_0$. But I'm having some troubles, anyone has a suggestion?

Asaf Karagila
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leluch_l8r4
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  • I imagine it's worth it to look at$$(\omega\uparrow\uparrow n)!=\left(\underbrace{\omega^{\dots^\omega}}_n\right)!,$$starting with the $n=2$ case, $\omega^\omega!$. – Akiva Weinberger Feb 15 '24 at 15:17
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    Defining $\Gamma(\lambda)=\prod\limits_{\beta < \lambda}\beta$ and $\lambda!=\Gamma(\lambda + 1)$ (I think this is somewhat standard in ordinal number arithmetic), then $\Gamma(\epsilon_0)=\epsilon_0$ and $\epsilon_0!=\Gamma(\epsilon_0)\cdot \epsilon_0 = \epsilon_0^2 > \epsilon_0.$ The result for $\Gamma(\epsilon_0)$ follows from $\epsilon_0 = {\omega}^{{\omega}^{\epsilon_0}}={\omega}^{{\omega}^{{\omega}^{\epsilon_0}}}$ and $\Gamma({\omega}^{{\omega}^{\alpha}})={\omega}^{{\omega}^{{\omega}^{\alpha}}}.$ (continued) – Dave L. Renfro Feb 15 '24 at 15:44
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    See the top of p. 318 of Sierpiński's Cardinal and Ordinal Numbers (1965 2nd edition) and the bottom of p. 212 of Accumulation functions on the ordinals by Rubin/Rubin (1971). I don't have time now to include all the details, so I'm putting this as a comment. – Dave L. Renfro Feb 15 '24 at 15:44
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    @DaveL.Renfro Answers don't have to contain all the details. Even the vaguest hint would be suitable for the answer section. Comments are specifically not for providing answers, the way you have done here. The actual comment box itself even says as much. – Arthur Feb 15 '24 at 16:06
  • Isn't it possible to prove $\alpha!\le \alpha^\alpha$ by induction on $\alpha$? – Hanul Jeon Feb 15 '24 at 19:37
  • @Arthur (and others): I'll get something written up in a couple of days, as I'm currently in the middle of a lot of time-consuming "day-job" work that I'm presently dealing with. I was able to get my earlier comments written quickly (5 minutes?) because I wrote (for myself) extensive notes on ordinal number arithmetic in the mid 1980s and I'm pretty familiar with both references I gave. I would like to give more details because as a general rule, I don't like to clutter my SE answers with relatively uninformative answers that are not especially worth preserving for later reference/citing. – Dave L. Renfro Feb 16 '24 at 09:19
  • @Arthur (and others): It's going to take me a bit longer. I have a good idea of what I want to write (a nice overview of transfinite sums, triangular ordinals, products, factorial ordinals; and a less detailed discussion about their higher-order operation versions), and I think it will be well worth having available in MSE, but it will take several more days, due to the amount of literature I am looking though and due to the care needed to ensure accuracy in all the symbolically-intricate aspects that are involved. – Dave L. Renfro Feb 19 '24 at 19:01
  • @DaveL.Renfro (and everyone above) wow! I'm still taking some time to think about the problem myself after shortly looking at the comments, so I haven't seen everything you mentioned in detail yet, but I'm extremely grateful to all of you for the very helpful contributions! – leluch_l8r4 Feb 19 '24 at 21:46
  • (Update) I've just finished the first 2 sections -- transfinite sums and triangular ordinals. Still a lot to go, but transfinite products and factorial ordinals should be quicker because the definitions are similar, and because how I've decided organize the exposition and what to include will be similar (much of this took a bit of thought and planning). Also, I've pretty much assembled and cited all the relevant literature I know about. (Had to make a trip to a nearby university library yesterday for 2 especially relevant books that I don't own, and I own quite a few set theory books.) – Dave L. Renfro Feb 22 '24 at 21:03
  • (Later Update) Still working. The product stuff took longer than expected, and many more details and observations and corrections made in the earlier parts, plus I've expanded my annotation of the references. Should finish products/factorials tomorrow, and I've already worked some on "exponential factorials" and "tetration factorials" that remain (just now made-up terms; maybe I'll use them). At this point, I think I'll finish later this week. As for length, it appears it's going to be even longer than this 3-part mathoverflow answer! – Dave L. Renfro Mar 03 '24 at 21:14
  • It seems exponential factorial is already a (sometimes) used term, although the order used there is backwards from what works for ordinal numbers and for ordinals one needs to use a bottom-up evaluation, not a top-down evaluation (e.g. see "Ordinal Tetration vs Usual Tetration" in this MSE answer). – Dave L. Renfro Mar 03 '24 at 21:29
  • (Still Later Update) Yep, still working. I've been sick roughly the past 5 days (quite a long time for me, but it's not COVID) and before that I began getting a bit "blah" with the topic, so progress has slowed, but still plodding along. Working on "exponential factorials" now (after having carefully discussed two types of continued exponentiation evaluations -- top-down and bottom-up; the former trivializes for infinite ordinals), then maybe a little with ordinal tetration, followed by some discussion of critical numbers for these operations (binary and transfinite versions). – Dave L. Renfro Mar 15 '24 at 11:10

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