10/23/2022, 07:29 PM
(10/23/2022, 06:57 PM)tommy1729 Wrote: I have no opinion about unicorns if you know what I mean.
If a definition is not given I will not say what it should be.
I'm not sure I understand what are you saying. Are you really saying that you have never seen a definition of indexed family of hyper-operations formal enough to have an opinion on the possibility of extending the argument of the indexing function, something that has often been called "rank", from natural numbers to larger sets? I remember you discussing the possibility of the half iterate of the superfuction operator, so I'm sure you know what here is meant by non-integer rank.
In other words, it seems to me that you claim that at this moment a precise definition of rank in the context of hyperoperations (see question 1) is lacking. Am I understanding your point?
Allow me to rephrase the other 3 questions to address your skepticism:
Quote:2') Is there some reason to be interested in the problem extending the \(n\) argument in expression like \(a\uparrow^n b\) from the natural number to non-integers?
3) What is, in your opinion, the main obstacle to solving the problem of non-integer ranks?
4) How do you see the mathematician of the future surpass this obstacle, if they ever manage to do it at all. In which field of mathematics do you see the key to the solution residing in?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)