Ivars Wrote:........................................
Sometimes I feel that with pentation .......................................
I can not see math as not being possible to close logically while allowing it to develop because of undetermined symbols like infinity, etc..
We are still working to clarify what happens up to the s=4 hyper-operation rank, i.e. at the tetration level. In particular, concerning the "priority to the right" business, we have:
(s=1, addition): a+(a+(a+a)) = a x 4 [bracketing not necessary, add. commutable, associable]
(s=2, multipl.): a.(a.(a.a)) = a ^ 4 [bracketing not necessary, mult. commutable, associable]
(s=3, expon.): a^(a^(a^a)) = a # 4 [bracketing indispensable, exp. non-commutable, non-assoc.]
(s=4, tetrat.): a#(a#(a#a)) = a § 4 [bracketing indispensable, tetr. non-commutable, non assoc.]
Symbol § is provisionally used here for "pentation". However, concerning the same priority business, for s=5 we must also have:
(s=5, pent.): a§(a§(a§a)) = a ç 4 [bracketing indispensable, tetr. non-commutable and non-associable]. Symbol ç means here "hexation". And ... so on.
The problem is that we have not fully analyzed yet the fourth rank level. We need to show that "sexp", "slog" and sroot" are smooth uniquely defined (1- or 2-valued) "functions", before proceeding further. Therefore, we are not ready to "sublime" up to the fifth rank.
Objects like "undeterminations", "infinitesimals" and "infinities" may strongly intervene in the analysis process, but I don't see them as really serius obstacles, if we pay the necessary attention, at this stage. I agree that the study of the hyper-operations, starting from the tetration rank might give new ideas also for a more complete approach to undeterminate and infinite magnitudes. We shall see!
GFR