I have been working hard on our Introduction to BEAF article, and it's coming along excellently. Any comments on the ordinal versions? I'm fairly convinced that legion arrays have BHO power, but I might be wrong. FB100Z • talk • contribs 23:41, December 6, 2013 (UTC)
BHO is only \(X\uparrow\uparrow X\text{&}X\text{&}n\). The limit is \(\psi(\Omega_\omega)\). Wythagoras (talk) 07:52, December 7, 2013 (UTC)
- I believe you, but what's the proof of this? (Then again, I've only managed to prove \(\varepsilon_0 = X \uparrow\uparrow X\) myself.) FB100Z • talk • contribs 20:27, December 7, 2013 (UTC)
- Anything? FB100Z • talk • contribs 22:03, December 9, 2013 (UTC)
- Sorry for not responding earlier.
- First, we shall prove \(X\uparrow\uparrow X2 = \varepsilon_1\).
- We know that \(X\uparrow\uparrow X2 = lim(X\uparrow\uparrow X,(X\uparrow\uparrow X)^{X\uparrow\uparrow X},(X\uparrow\uparrow X)^{(X\uparrow\uparrow X)^{X\uparrow\uparrow X}}...)\) and \(\varepsilon_1 = lim(\varepsilon_0,\varepsilon_0^{\varepsilon_0},\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}...)\).
- Since \(\varepsilon_0 = X \uparrow\uparrow X\), \(\varepsilon_1 = X \uparrow\uparrow X2\). It is easy to extend this proof to \(\varepsilon_k = X \uparrow\uparrow X(k+1)\), and then proving that \(X\uparrow\uparrow\uparrow X = \zeta_0\) shouldn't be that hard too.
- But after that, it gets somewhat complicated, but with a proof \(\{X,X,n+1\}\text{&}n=\varphi(n,0)\) I can do it. Wythagoras (talk) 06:27, December 10, 2013 (UTC)
- Hold up. In the article, I defined \(\omega \uparrow\uparrow (\omega + 1) = \varepsilon_1\). Can you give me a reason why \(\omega \uparrow\uparrow \omega 2\) is better? FB100Z • talk • contribs 06:31, December 10, 2013 (UTC)
- \(\omega\uparrow\uparrow(\omega+1)=\omega^{\omega\uparrow\uparrow\omega+1}=\omega^{\varepsilon_0+1}\). \(\omega\uparrow\uparrow(\omega+n)\) are given by adding more \(\omega\)'s, and in the limit they give \(\varepsilon_1=\omega \uparrow\uparrow \omega 2\). LittlePeng9 (talk) 14:34, December 10, 2013 (UTC)
- But after that, it gets somewhat complicated, but with a proof \(\{X,X,n+1\}\text{&}n=\varphi(n,0)\) I can do it. Wythagoras (talk) 06:27, December 10, 2013 (UTC)
Sorry for being stubborn, but I'm still not convinced. It seems that there are multiple interpretations of what arrow notation and BEAF does for transfinites — one is \(\varepsilon_1 = \omega \uparrow\uparrow (\omega + 1)\), and the other is \(\varepsilon_1 = \omega \uparrow\uparrow \omega 2\). FB100Z • talk • contribs 23:00, December 10, 2013 (UTC)
- And by Saibian's variant, . By his definition, and unlike other notations, has exactly entries. Ikosarakt1 (talk ^ contribs) 20:34, July 14, 2014 (UTC)
