User blog:Ubersketch/Regarding eventually computable functions

In googology, the vast majority of functions are computable. This is likely because of their comparability and computability by immortal human minds given enough time to do the necessary calculations. Lying beyond the threshold of computability, exists the ITTM computable functions (I like to call them infinitely computable), which can be calculated by a turing machine (with 3 tapes) that can run in transfinite time and halt. Within here lies the majority of uncomputable functions, including Rayo(n) and BB(n). Even farther still, lie the eventually computable functions, which can be calculated by a turing machine (with 3 tapes) that can run in transfinite time and reach a stabilized value, even if it doesn't halt. As far as I know, nobody has gone far enough to create any functions that lie in here, apart from the obvious cases using the FGH and the limit of the eventually computable ordinals. Anything further than here is a vast landscape, labelled on maps as "here be dragons." To create a function beyond eventual computability, you could try using stable ordinals in the FGH. Stable ordinals use a generalization of the admissibility of the limit of computability, the Church-Kleene ordinal. There are also ordinal notations for these levels. Kleene O designates computable ordinals, Klev O+ designates infinitely computable ordinals, Klev O++ designates eventually computable ordinals. So far there has been no rigorous notation for countable ordinals beyond eventual computability. There is also a concept of accidental computability, where an ordinal appears on the tape but isn't the actual result. Perhaps we can make a notation for accidental computability.