[bvssvni]

One thing I would like point out with Gödel's incompleteness theorems, is that there are different notions of provability. Gödel uses the notion of "provability" you get from Provability Logic, which is a modal logic where you introduce terms from e.g. Sequent Calculus.

Recently I found another notion of "provability" where Löb's axiom, required in Provability Logic, is absurd. It turns out that this notion fits better with Intuitionistic Propositional Logic than Provability Logic. This allows integrating the meta-language into object-language. This is pretty recent, so I think we still have much to learn about the foundations of mathematics.

[drdeca]

This makes it kind of sound as if Löb's axiom is used in the proof of Gödel's incompleteness theorems, but Gödel was working in (I think) PA, not in Provability Logic?

I guess you just meant "the notion of provability is the same as the one that would later be described in Provability logic" ?

I viewed the page you linked, but I don't see anywhere where you describe the alternate notion of "provability" you have in mind.

"Provability" in the sense of Gödel's incompleteness theorems, is just, "There exists a proof of it in [the chosen system of axioms + rules of inference]", yes? I don't see how any other notion would be fitting of the name "provability". "provable" , "possible to prove", "a proof exists", all the same thing?

Oh, I guess, if you want to use "there exists" in a specifically constructive/intuitionistic way? (so that to prove "[]P" you would have to show that you can produce a proof of P?)

I think I've seen other people try to do away with the need for a meta-language / make it the same as the object-language, and, I think this generally ends up being inconsistent in the attempts I've seen?

edit: I see you have written 'Self quality a ~~ a is equivalent to ~a, which is called a "qubit".'

I don't know quite what you mean by this, but, _please_ do not call this a qubit, unless you literally mean something whose value is a vector in a 2d Hilbert space.

[bvssvni]

> I guess you just meant "the notion of provability is the same as the one that would later be described in Provability logic" ?

yes

> I viewed the page you linked, but I don't see anywhere where you describe the alternate notion of "provability" you have in mind.

See section "HOOO EP"

>Oh, I guess, if you want to use "there exists" in a specifically constructive/intuitionistic way? (so that to prove "[]P" you would have to show that you can produce a proof of P?)

This would be `|- p` implies `□p`, which is the N axiom in modal logic (used by Provability Logic).

In Provability Logic, you can't prove `□false => false`. If you can prove this, then Löb's axiom implies `false`, hence absurd. `□p => p` for all `p` is the T axiom in modal logic. In IPL, if have `true |- false`, then naturally you can prove `false`. So, you can only prove `□false => false` if you already have an inconsistent theory.

Provability Logic is biased toward languages that assume that the theory is consistent. However, if you want to treat provability from a perspective of any possible theory, then favoring consistency at meta-level is subjective.

> I think I've seen other people try to do away with the need for a meta-language / make it the same as the object-language, and, I think this generally ends up being inconsistent in the attempts I've seen?

I have given this some thought before and I think it is based on fix-point results of predicates of one argument in Provability Logic. For example, in Gödel's proof, he needs to encode a predicate without arguments in order to create the Gödel sentence. In a language without such fix-points, this might not be a problem.

> I don't know quite what you mean by this, but, _please_ do not call this a qubit, unless you literally mean something whose value is a vector in a 2d Hilbert space.

The name "qubit" comes from the classical model, where you generate a random truth table using the input bit vector as seed. So, the proposition is in super-position of all propositions and hence behaves like a "qubit" in a classical approximation of a quantum circuit.

[deltasepsilon]

I'm attempting to understand you, so:

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?

How so, and/or, what are the axioms, or derived properties, of this new notion of provability you have uncovered?

[bvssvni]

> 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.

[layer8]

> Recently I found another notion of "provability"

Any pointers to that alternative notion?

[bvssvni]

I'm implementing it in this project: https://crates.io/crates/hooo