# Can, "This problem is unsolvable," be used to formulate the first incompleteness theorem in erotetic logic specifically?

Assumptions/definitions: the Gödel sentence is informally equivalent to, "This sentence can't be proved in system X," where X is appropriately specified. Since that sentence can itself be created inside such a system, then, the system can be seen as giving rise to self-unprovable sentences, such that the system is not able to completely prove itself (when taken as a conjunction of all the sentences that can be given in and from the system).

Now, erotetic logic is centered on questions, but, "This question cannot be answered in system Y," is ill-stated (since it is not a question). The concept of (abstract) problems is quite erotetic in character, but we often use the word "problem" in such a way as indicates a factive aspect/component to what problems are. So we assume (or hope) that, "This problem is unsolvable in system Y," is either well-stated or ill-stated in a different way. If it is still ill-stated, my first suspect in the case would be representing such a sentence as a problem: how is it a problem-to-solve (by some process)? Can a sentence just declare of itself that it is problematic "by stipulative definition"?

If, "This problem is unsolvable in system Y," goes through, and retains a deep erotetic flavor while existing also in substantively factive terms, then is it possible to state the first incompleteness theorem in these same terms? I'd be hard pressed to claim that I knew what erotetic Gödel codes would look like (if you will), but as an outsider looking in, I wouldn't expect that the coding would be prohibitively difficult.

Upshot? The Gödel sentence and the attendant incompleteness issues have been noted to be similar to the knower-paradox sentence and Fitch's paradox of knowability. But if axioms are knowable, then knowledge goes beyond proof, and at any rate, "to solve a problem" seems expansively inclusive over "to prove a point" in that proofs are solutions to problems (of validity/inference) but there are problems with non-proof solutions (or with both provable and unprovable, but either way knowable, solutions, perhaps). So instead of allowing for a newfangled image of the incompleteness problem, does the unsolvable sentence emerge just as a shadow of the knower-paradox sentence?

EDIT: to me, the unsolvable sentence seems potentially true. If it can be a problem by defining itself that way, it might be yet given to define itself so obtusely that it is, however, unsolvable (in that it lacks the "word problem details," so to say, that we'd need to solve it). So it is an unsolvable problem period, arguably, not just in some system or another. But this would favor the analogy with the unknown sentence rather than the unprovable one, maybe.

• "This sentence can't be proved in system X" cannot be expressed inside X when X is Peano arithmetic, it lacks self-referential devices. This is why Gödel had to use the numbering to produce a modified sentence. Smullyan constructed languages where such self-reference is directly possible. If you want an erotetic formulation what is wrong with "is this problem unsolvable?" Btw, the negation of Gödel sentence is equally unprovable, but they cannot both be "potentially true". Your "potential truth" has to be based on more than just unsolvability. Mar 17 at 21:46
• @Conifold I don't know why I didn't think about the unsolvable problem manifesting as both a question and an assertion, and I don't know what the implications will be, but I should look into the matter. I don't know that I'd end up with an erotetic counterpart of the unprovable sentence instead of the unknowable one, though... By "negation of the Gödel sentence sentence" do you mean, "This sentence is provable?" (I can hear an intuitionist raven screeching, "Not nevermore!" at me, haha!) But so, "This problem is solvable," seems like it wouldn't line up with that: Mar 18 at 13:10
• "This sentence is provable," seems false (the indexical closure of the sentence seems to prevent inferring it from any intelligible premises), but, "This problem is solvable," seems trivially true, in the sense of having a trivial solution (itself, in being stated, even). At least, those are my impressions right now. Mar 18 at 13:11
• No, I mean the negation of the actual Gödel sentence in Peano arithmetic. Its identification with "this sentence is unprovable" is highly misleading as it mixes language with meta-language. After initially using it casually to explain the idea informally, Gödel dropped it when Zermelo, Wittgenstein and others misinterpreted the incompleteness theorem as a result. The point is that there is no intrinsic connection between Gödel-like sentences and "this is provable/unprovable" which could determine their "potential truth". It depends on external choice of interpretation. Mar 18 at 14:12
• @Conifold without appealing to self-reference, then, can one formulate a Gödel-like sentence based on solvability instead of provability, either to the same or even more general effect? Or does generalizing it lend itself to a version of Fitch's knowability paradox instead? (Can the unknowable sentence be reformulated so as to not be self-referential?) Mar 18 at 14:16