The basic constraint is that if you want to prove something based on purely logical statements, you need to express the problem precisely enough for pure logic, that is without any wiggle room open to vauge human interpretation, and the notions you use have precise meaning. If not, your problem can't be solved by pure logic, but only by more wishy washy handwavy arguments, from which you can't derive any insights with logical certainty.

You might be able to derive philosophical or empirical meaning from an interpretation of some non-precise statement, but you shouldn't hope that such an interpretation is a valid mathematical theorem.

> The basic constraint is that if you want to prove something based on purely logical statements, you need to express the problem precisely enough for pure logic, that is without any wiggle room open to vauge human interpretation, and the notions you use have precise meaning. If not, your problem can't be solved by pure logic, but only by more wishy washy handwavy arguments, from which you can't derive any insights with logical certainty.

I think when you say "pure", you actually mean "formal" or "symbolic" or "mathematical". Many believe that there is much more to logic than a mere formalism – although that itself is a major point of contention in the philosophy of logic.

> You might be able to derive philosophical or empirical meaning from an interpretation of some non-precise statement, but you shouldn't hope that such an interpretation is a valid mathematical theorem.

Berry's paradox may not be a mathematical paradox – but that doesn't show it is not a paradox. And, even if it is not a mathematical paradox, it nonetheless made a valuable contribution to mathematics, by inspiring the construction of formal analogues, some of which turned out not to be paradoxical (but nonetheless interesting). However, I don't think it necessarily follows, that just because those formal analogues turned out not to be actual paradoxes, that its natural language version cannot be a real paradox. And, some argue there are formal analogues which are genuine paradoxes – see for example https://ojs.victoria.ac.nz/ajl/article/download/4972/4634/73...

> some argue there are formal analogues which are genuine paradoxes...

Isn't this getting a bit circular, as it seems to be using the presumed existence of a formal analogue to justify the informal Berry's paradox? If it has a paradoxical formal analogue, then it does not stand as an example of a useful yet unavoidably informal paradox.

Also, given that there seem to be more than one plausible formal analogue, not all of which are paradoxes, does the dispute not just re-form around which is the real formal analogue? That's what seems to be happening in the exchange of papers from which your example was taken.

>How can you say something “does not mean anything” simply because it cannot be formalised? Since when is formalisability a prerequisite for meaningfulness?

I'm wondering then, what you think is the interest of this theorem?

The goal of Gödel was to create a metamathematics. You need to know what is a paradox of the theory or a paradox or not to know if it is correct. If you have a contradiction, you cannot believe anything that the theory proves. So the whole thing is ultimately about separating mathematical paradox from "other types" of paradoxes.

> I'm wondering then, what you think is the interest of this theorem?

Well, it was a profound discovery about the limits of consistent formal systems, and of course it is interesting in that light. But, viewing that as profoundly interesting does not require one to adopt your apparent position that only the formalisable is "real".

> If you have a contradiction, you cannot believe anything that the theory proves.

Only with classical logic. With paraconsistent logic, you can believe things proven from inconsistent axioms. Classical logic includes the principle of explosion (ex contradictione quodlibet, from a contradiction anything follows); so inconsistent theories immediately collapse into trivial ones, and trivial theories are of course utterly useless. But since paraconsistent logic rejects the principle of explosion, inconsistent theories do not collapse into triviality, and thus may be useful in spite of their inconsistency.

Can you explain to me what is the "realness" of Berry's paradox?

Classical logic is a poor model of natural language and informal human reasoning. Paraconsistent logics are much more accurate formal models of informal human reasoning and natural language, and much closer to them in power.

One argument for this: material implication is a poor model of how implication works in natural language, as the well-known paradoxes of material implication show. With modal logic, we can introduce strict implication, which performs better, but still falls short as an accurate model of natural language implication, as shown by the paradoxes of strict implication. Relevant logics (aka relevance logics) avoid the paradoxes of strict implication as well. But relevant logics are also paraconsistent.

Using an appropriate paraconsistent logic, Berry's paradox can be turned into a formal paradox, and such a formal paradox can be viewed as a straight-forward formalisation of the informal Berry's paradox.

Attempts to formalise Berry's paradox using classical logic fail to produce a paradox, because classical logic lacks sufficient self-referential power – classical logic cannot cope with inconsistency, so classical formal systems must have their powers of self-reference neutered – compared to natural language – to prevent self-referential paradoxes – and those restrictions also prevent a formal Berry's paradox from existing. Paraconsistent logic is far better at handling inconsistency, so it does not have to restrict self-reference, and becomes much closer to natural language in power – and Berry's paradox becomes a formal paradox as well as an informal one.

So, given the above – in what sense is Berry's paradox not "real"?