User blog comment:Planterobloon/How do you compare huge numbers to each other?/@comment-11227630-20170812012304/@comment-30754445-20170813043923

Obviously, which is why Goodstein's function would be a pretty bad "standard notation" to use when comparing big numbers. And even with such notations, using inequalities often complicates things needlessly.

One of the few places where I found inequalities to be actually useful, was in finding the thresholds of the Mashimo Scale. Doing a straightforward calculation of M(n) seems much more cumbersome than bounding it with increasingly tighter bounds.

Another place where inequalities are necessary, is when a big number solves an actual mathematical question which we don't yet know the answer to. With things like BB(5) or TREE(3) or SCG(13) or the actual answer to the "Graham Number" question, we have no choice but to use inequalities (I still get kicks from the fact that BB(5) is 'too large to be calculated' yet it is probably exactly equal to 4098 :-))