"Someone who denies is a denialist." I deny that moon is cheese, hence I guess I'm a "denialist", too. Are you? The term carries a negative tone and is used to label people. We should be able to offer different perspectives and even controversial views without being labeled. Many great ideas in science were controversial when first presented. Label the claim if you will, not the human behind.

Here are some pointers. math.stackexchange.com/questions/24662/… math.stackexchange.com/questions/3020186/…

@JoelDavidHamkins Fixed, thank you.

Very interesting pointer, thank you. It seems you have worked on the same core problem but at different tool set. I need to study this more deeply.

@FedorPakhomov Yep, the existence itself is clear. We could, for example, use the fast-growing hierarchy and define mentioned subrecursive hierarchy as closures over the PR and a chosen $f_\alpha$ for $\alpha<\epsilon_0$. However, I'm interested if we can define a hierarchy without any notion to ordinals, in (kind of) more "constructive" way just using the higher order recursion schema, but I have not been able to locate any references (only those around lambda calculus that employ different tools).

@FedorPakhomov Yes, I'm targeting "pure number-theoretical" setup here, thus the lambda abstraction is somewhat out of scope. The core idea is to reach a hierarchy like that of Grzegorczyk's within the class PR, but within the class of functions provable total in PA. I remember reading about the combinators earlier, I need to revisit that. Thanks for the pointer.

If you define "natural" to be something like (Kalmar) elementary, you might be interested in the Ritchie-Cobham property of elementary functions. It says that a function is elementary iff it is computable in elementary time. From this perspective one needs something non-elementary to break through the elementary growth rates ceiling. (not sure if this of interest, but might be a sort of metatheorem you are looking for)