User blog comment:Chronolegends/improved Egg Notation - Analysis/@comment-28606698-20170427070833/@comment-27513631-20170429080438

If you're going up to \(\varepsilon_0\), why use the fgh when you can use the hardy hierarchy and shave off a couple of rules? Besides tradition/building up from a simpler notation.

Also, it's quite easy to represent any code with just ones and zeroes, as in iota and not (https://en.m.wikipedia.org/wiki/Iota_and_Jot). Unfortunately, these are turing-complete. A better way to get extremely simple systems with incredible strength would be type theory, as in loader's number, but they aren't hierarchically built.