> Are you saying that Löb's axiom, which states that the provability of "the provability of p implies p" implies the provability of p, necessarily prejudices some implicit assumption of consistency to the meta-language?

Yes. One way to put it: Provability Logic is balancing on a knife-edge. It works, but just barely. However, you can turn it around and say the new notion is balancing on a knife-edge by requiring DAG (Directed Acyclic Graph) at meta-level. They way I see it, is that both approaches have implicit assumptions and you have to trade one with another.

I am working on an implementation of the new notion of provability (https://crates.io/crates/hooo), after finding the axioms last year (it took several months):

    pow_lift : a^b -> (a^b)^c

    tauto_hooo_imply : (a => b)^c -> (a^c => b^c)^true

    tauto_hooo_or : (a | b)^c -> (a^c | b^c)^true

As a modal logic the difference is surprisingly small, by swapping Löb's axiom with T. `tauto_hooo_imply` is slightly stronger than K.

The major difference is that `|- p` equals `p^true` instead of implying, if you treat `|-` as internal. If you treat it as external, then you can think of it as N + T.

I needed this theory to handle reasoning about tautological congruent operators.

However, once you have this, you can perfectly reason about various modal logical theories by keeping separate modality operators, including Provability Logic, e.g. `modal_n_to : N' -> all(a^true => □a)`.

So, it is not a tradeoff that loses Provability Logic. Instead, you get a "finalized" IPL for exponential propositions. This is why I think of as a natural way of extending IPL with some notion of provability.