03/29/2015, 03:51 PM
Ah thanks... I'm a newbie now so I can't judge you old ideas. Actually I don't even understand how Jay came at that formula and reading bo seems to me that the equation is not proved yet. Anyways, assuming it is correct, it looks amazing!
It wold be really amazing to find that the basechange of pentation is linked with something that is in some sense "before the addition". But the question is ... in wich sense is "before the addition"?
In fact, as long as we want the "Akermann" recurrence (call it mother law or how you like) to hold across the ranks (negative and non-natural as well) we are still with the problem of the successor function being a fixpoint of the subfunction operator... aka all the negative ranks are the successor (as Bo noted long ago, as Jmsnxn proved and as you noted long ago in JmsNxn's post about extending Ackermann).
Also if we drop the "Ackermann's recursion" (\( H_{s+1}\circ S=H_{s}\circ H_{s+1} \)) then we just have to move on other Hyperoperations families... like Bennet's one... but imho Ackermann's recursion is too much important and fascinating, if you think about it is that recursion that give us tetration and pentation, ..., and \( s{\rm -ation} \), while in other families like the balanced, the bennet or the right-distributive (lower Hos) we never reach them.
It wold be really amazing to find that the basechange of pentation is linked with something that is in some sense "before the addition". But the question is ... in wich sense is "before the addition"?
In fact, as long as we want the "Akermann" recurrence (call it mother law or how you like) to hold across the ranks (negative and non-natural as well) we are still with the problem of the successor function being a fixpoint of the subfunction operator... aka all the negative ranks are the successor (as Bo noted long ago, as Jmsnxn proved and as you noted long ago in JmsNxn's post about extending Ackermann).
Also if we drop the "Ackermann's recursion" (\( H_{s+1}\circ S=H_{s}\circ H_{s+1} \)) then we just have to move on other Hyperoperations families... like Bennet's one... but imho Ackermann's recursion is too much important and fascinating, if you think about it is that recursion that give us tetration and pentation, ..., and \( s{\rm -ation} \), while in other families like the balanced, the bennet or the right-distributive (lower Hos) we never reach them.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
