06/30/2022, 11:41 PM
(06/28/2022, 08:33 AM)Catullus Wrote:(09/01/2019, 04:34 AM)VSO Wrote: Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?Although, each level of the Ackermann function is primitive resursive. The entire Ackermann function itself can not be de-recursed.
This comment doesn't address VSO's question. Ackermann function do not coincides with hyperoperations. Secondly, hyperoperations need not to be defined recursively. Third point: ackermann function, goodstein function (natural hyperoperations), if considered as functions over the natural numbers are recursive. Are defined recursively.
It is not fully clear what do you mean by de-recursed. Anyways, if the question is if it's possible to give an analytical, non recursive, expression to the ackermann function/hyperoperations, the answer is yes.
JmsNxn gave one expression for that. The real question is if that analytical representation has desired properties: does it satifies a functional equation? Is it smooth? Holomorphic?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
