Talk:Buchholz hydra

Any values of the first few terms? Also, can anyone program it at Turing machine? It would be a big breakthrough in googology, because with this function we can easily beat even the number proposed by Chris Bird at the end of his paper. Ikosarakt1 (talk ^ contribs) 13:49, April 10, 2013 (UTC)

Well, (+,0) dies after one cut, so BH(0) = 1. (+,0,\(\omega\)) goes to (+, 0, 2), which goes to (+, 0, 1, 0), which goes to (+, 0, (1, 1, 1)). It gets quite complicated after that, so even BH(1) is hard to determine, although it doesn't surpass the Bachmann-Howard ordinal. Deedlit11 (talk) 14:13, April 10, 2013 (UTC)
 * I thought TFB was greater than Bachmann-Howard? FB100Z &bull; talk &bull; contribs 01:17, April 11, 2013 (UTC)


 * It is. The strength of trees labelled with 0's and 1's is the Bachmann-Howard ordinal, so BH(1) can be at most f_BHO(n) for a small n. Deedlit11 (talk) 01:41, April 11, 2013 (UTC)

I haven't studied this much, but what if we allow other ordinals?


 * 1) The first rule is unchanged.
 * 2) If node we chose is a successor ordinal \(\upsilon + 1\), we go down the tree looking for a node \(b\) with label \(\phi \leq \upsilon\), etc.
 * 3) If the node we chose has a limit ordinal \(\upsilon\) for its label, we relabel it with \(\upsilon[n + 1]\).

How much would this benefit us? FB100Z &bull; talk &bull; contribs 19:53, April 10, 2013 (UTC)


 * I believe that, using ordinals up to \(\alpha\), we can get a growth rate of \(f_{\psi_0(\varepsilon_{\Omega_{\alpha}+1})}(n)\). If we use trees as labels for trees, then use those trees as labels for even larger trees, etc., we reach a limit of \(f_{\psi_0(\Omega_{\Omega})}\). Deedlit11 (talk) 20:01, April 10, 2013 (UTC)
 * Actually, if I recall correctly, limit of \(\psi\) function is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), TFB ordinal, so we probably need stronger notation for that. I also had this idea, although I have no idea about its strength. LittlePeng9 (talk) 20:11, April 10, 2013 (UTC)
 * Yes. TFB marks the limit of the psi-0 function. We need a new function to go any further. FB100Z &bull; talk &bull; contribs 01:01, April 11, 2013 (UTC)
 * Again, why TFB ordinal is limit for \(\psi_0(\alpha)\) function? Is it not possible to have ordinal, say, \(\psi_0(\zeta_{\Omega_\omega+1})\)? Ikosarakt1 (talk ^ contribs) 10:16, April 11, 2013 (UTC)
 * \(\psi_0\) function is said to be eventually constant. It means that ordinal you wrote is no greater than TFB ordinal. I think every collapsing function asks for smallest not in a given set. TFB isn't in such set for any ordinal, and this is connected to fact that \(\varepsilon_{\Omega_\omega +1}=\Omega_\omega^{\Omega_\omega^{...}}\), so using exponentiation we can't reach it and then apply \(\psi\) function to it. LittlePeng9 (talk) 15:47, April 11, 2013 (UTC)


 * It's straightforward to extend the notation - just add more cardinals of the form \(\Omega_{\alpha}\) (and add corresponding functions \(\psi_{\alpha}(\beta)\)). But the stronger notations that I have seen use cardinals as subscripts for \(\psi\), so I guess I should say that the Buchholz hydras indexed with ordinals up to \(\alpha\) have a growth rate of \(f_{\psi_{\Omega_1}(\varepsilon_{\Omega_{\alpha}+1})}(n)\). Deedlit11 (talk) 13:24, April 12, 2013 (UTC)


 * Given that it has three rules &mdash; zero, successors, and limits &mdash; it looks suspiciously like an ordinal hierarchy... FB100Z &bull; talk &bull; contribs 01:01, April 11, 2013 (UTC)


 * It sure does! The Buchholz hydras closely match an ordinal notation for ordinals up to \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\). In fact, if you read Buchholz's paper, he exploits this relationship to prove that the termination of Buchholz hydras cannot be proven in \(\Pi^1_1 \text{- CA + BI}\). Note that the reduction of Buchholz hydras closely match up with taking fundamental sequences of ordinals in the ordinal notation, except that the rule for when you chop off a head with label u+1 isn't quite right - the rule only appends one subtree, whereas to be a proper fundamental sequence it should append n subtrees, one on top of another. But Buchholz shows it doesn't fundamentally affect the growth rate. Deedlit11 (talk) 13:38, April 12, 2013 (UTC)

Implementation
I finally got around to implementing this beast (pun intended). Are these first few steps of BH(3) correct?

(+(0(ω))) (+(0(2))) (+(0(1(0)))) (+(0(1)(1)(1)(1))) (+(0(1)(1)(1)(0(1)(1)(1)(0)))) (+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1)))) (+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1)(0))))) (+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))))) (+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(0(1)(0)))))) (+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))))))

FB100Z &bull; talk &bull; contribs 20:45, April 16, 2013 (UTC)

Looks right to me. Deedlit11 (talk) 20:50, April 16, 2013 (UTC)