So, since definitions are circular and the scope of natural languages take place in the world itself, then do you think that there's anything that can be said about Gödel's incompleteness theorems, within such a context?

I'll bite. Non-formal language is descriptive, as such useful - "true" - wrt certain criteria, and as such never itself the truth. And the same for formal/technical language, the difference being that those users usually do not worry about truth.

All this contained in Bacon's "putting nature to the question." What nature returns is an answer to the question.

I'll bite. Non-formal language is descriptive, as such useful - "true" - wrt certain criteria, and as such never itself the truth. And the same for formal/technical language, the efference being that those users usually do not worry about truth. — tim wood

So, I think you are right about the centrality of truth in formal systems with axioms. Therefore, do you think that the very fact that we have circular definitions in our natural language denies what Gödel's incompleteness theorems can say about such languages?

I don't think Godel's has anything to do with natural languages. What is your target, here?

I don't think Godel's has anything to do with natural languages. — tim wood

Why is that? What makes his theorems unapplicable to natural languages?

I didn't say inapplicable. Applicability and application are in the eye of the beholder.

Godel-Rosser is:

If T is a consistent formal theory adequate for arithmetic, then T is incomplete.

The steps in the proof depend on T being a formal theory.

What would "extend the import to non-formal languages" mean?

What would "extend the import to non-formal languages" mean? — TonesInDeepFreeze

As the topic progressed, I simply will rephrase the OP to look, as the following:

Are Gödel's incompleteness theorems applicable to non-formal languages?

If so, then the question follows, if it doesn't then it does not. Therefore, I think the need for that condition to be met is a sine qua non for Gödel's incompleteness theorems to have anything to do with non-formal languages, such as natural languages.

Thank you.

Some people believe that Godel-Rosser has implications not confined to mathematics and questions in the philosophy of mathematics. They argue that Godel-Rosser pertains to questions in epistemology, ontology, and even other subjects. But I don't know what you mean by 'applicable to non-formal languages'.

An excellent book that discusses arguments about Godel-Rosser outside of mathematics and philosophy of mathematics is 'Godel's Theorems' by Torkel Franzen.

At the sake of sounding incoherent, I just wanted to provide my rationale for asking (what I perceive) a fundamental question about formal and non-formal languages where Gödel's incompleteness theorems can apply.

What I suppose is the fundamental question is at which point does Gödel's incompleteness theorems manifest their truth in a natural language, if applicable?

But I don't know what you mean by 'applicable to non-formal languages'. — TonesInDeepFreeze

I suppose this is a statement more apt for discussion under the guise of T-schema's and Tarski's undefinability theorem, no?

An excellent book that discusses arguments about Godel-Rosser outside of mathematics and philosophy of mathematics is 'Godel's Theorems' by Torkel Franzen. — TonesInDeepFreeze

Yes, thanks for this. I'll be sure to give it a look.

Yes, Tarski was very much concerned with both formal and natural languages.

Very marginal, imho. Read Philosophical Investigations instead.

If it were possible to extend the import of Gödel's incompleteness theorems on non-formal languages, then what would they be? — Shawn

We could possibly relax the definition of the predicate isProvable(n) to isRational(n). Say that if the majority of the observers believe that a natural-language sentence is rational, then it is.

In that case, by using the diagonal lemma, we can assert that:

In natural language, there exists a false sentence that is rational or a true sentence that is not rational, or both.

Next, for the same reasons as for Godel's theorem, we can safely assert that the overwhelming majority of true sentences in natural language are not rational.

Very marginal, imho. Read Philosophical Investigations instead. — 180 Proof

Yes, I have something to say about how Wittgenstein approached Gödel's incompleteness theorems as logical paradoxes. Namely, because Wittgenstein took the world as the totality of facts, not things, then the sum total of facts operating in the state of affairs of the world, necessitates that logic takes care of itself.

There's no Neo or glitch in the system itself...

Everything is a tautology...

We could possibly relax the definition of the predicate isProvable(n) to isRational(n). Say that if the majority of the observers believe that a natural-language sentence is rational, then it is. — Tarskian

Yet, is rationality truth-apt, as you've defined it? At least if it's epistemologically denoted, then these observational sentences are truth-apt, no?

Yet, is rationality truth-apt, as you've defined it? At least if it's epistemologically denoted, then these observational sentences are truth-apt, no? — Shawn

Yes, it is limited to logic sentences expressed in natural language.

-- It is raining now. -> part of the universe of sentences under consideration for rational incompleteness.

-- Eat with fork and knife! -> not part of this universe.

Hence, it is only about sentences that can have a truth value.

Most true sentences will turn out to be irrational, if only, for model-theoretical reasons. They will be true in one model/interpretation but not in one or more other ones. Rationality has the same problem as provability.