User blog comment:MilkyWay90/Yet another attempt at making a fast-growing function/@comment-35434595-20180709002245/@comment-30754445-20180709032050

That's a tiny subset of the language of Peano arithmetic, and the resulting function would be quite weak. In fact, it would be on the order of n^n, because the biggest number you could write in such a language would be:

n*n*n*n*n...

In actual Peano arithmetic, you also have axioms and logical rules that enable you to write proofs.So you need lots and lots of symbols for things like "for all numbers n", "there exist a number n", "if/then", "and", "or" and so forth. And it is these concepts that allow you to define huge numbers with a "reasonable"(*) amount of symbols.

(*) "reasonable" is a relative concept here. You'll still probably need hundreds of symbols to define anything googological in PA. This is why Rayo's number is defined as Rayo(10^100) and not (say) Rayo(99). Because these kinds of functions usually have a very shaky start.