Do Gödel sentences (even indirectly) assert only their own unprovability?

Question

Sometimes the basic Gödel sentence is said to mean something like, "This sentence is unprovable in system F." Perhaps more correctly, it is sometimes said to mean something like, "There is a sentence unprovable in system F," with the caveat that that sentence itself turns out to satisfy that very claim.

But the Gödel sentence isn't absolutely unprovable, is it? Because we will note that it can be proven in a higher system; it's just that the higher system will go on to have its own Gödel sentence, and so on and on. If a Gödel sentence were really supposed to be self-referential, wouldn't it be doubly so, as something like both, "I am unprovable in system F," but also, "I am provable in system F⁺"?

C.f. what the SEP article on the incompleteness theorems says:

1. In the section on diagonalization and "self-reference": It is often said that given a property denoted by A(x), the sentence D is a self-referential sentence which “says of itself” that it has the property A. Such figures of speech may be heuristically useful, but they are also easily misleading and suggest too much.

2. In the next section: In informal explanations of the first incompleteness theorem, it is often said that the Gödel sentence G_F “says of itself that it is not provable”. Such imprecise statements, however, should be taken at least with a grain of salt. There are a number of reasons to conclude that, at least in general, Gödel sentences do not really say anything substantial about themselves (Milne 2007 is a careful analysis of such issues); for example, as was previously noted in the case of the Diagonalization Lemma, one is usually operating here with mere material equivalences.

3. In a much later section: Heuristically, one may view the Gödel sentence G_F as expressing its own unprovability—saying “I am not provable”—though, as was already emphasized, such claims should be taken with a grain of salt. Leon Henkin put forward the question whether the sentence expressing its own provability (“I am provable”) is true or false, and provable or not (Henkin 1952). Georg Kreisel soon pointed out that this depends vitally on how provability is expressed; with different choices, one gets opposite answers (Kreisel 1953).

Answer 1

Here's one way of understanding this. If you take the sentence "this sentence is not provable", it is objectionable in just the same way that the liar paradox is. It refers to itself in a non-well-founded fashion. Tarski's response is to introduce a hierarchy of object language, metalanguage, metametalanguage, etc. A simple way to eliminate both the liar paradox and the difficulties attending "this sentence is not provable" is to impose a rule that sentences may not violate the hierarchy.

It helps when illustrating the distinction to use a different language between the object language and metalanguage. Consider:

La neige est noir.

The above is a sentence that I have selected to be in my object language. It is in French.

"La neige est noir" is false.

The above is a metalevel sentence in English that says of the object language sentence that it is false. Both of these are unproblematic: they do not refer to themselves. Now consider:

This sentence is false. This sentence is not provable.

These sentences violate the Tarskian hierarchy by trying to be both an object language sentence and a statement about an object language sentence. One of the simplest responses to such sentences is just to say they are not permitted, because of this violation. There are many other sophisticated ways of handling the liar paradox, but this is not the place for that.

Now Gödel comes along and does something ingenious. He shows how we can take a formal system of arithmetic, with its precisely defined language and provability relation, and formulate an arithmetical predicate in the object language that is satisfied by a number n if and only if n is the Gödel number of a provable sentence. This arithmetical predicate is abbreviated Bew(n) but it is a formula in arithmetic that can be written out using the language of first order logic.

If G is a sentence in the object language, it would not do to write:

G if and only if ⊬ G

since that violates the hierarchy. G is in the object language and ⊬ G is in the metalanguage. But thanks to Gödel we can instead write

G ↔ ¬Bew(⌜G⌝)

which lies entirely within the object language. (Where ↔ is the material biconditional and the corner quotes indicate Gödel number.) Gödel's proof shows that for formal systems that are consistent, recursively axiomatisable and sufficiently strong, there exists such a G. A Gödel sentence G is long and complicated, but it is not self-referential and it is not self-contradictory. For an example of what one looks like, see this answer on MathSE.

So, when your quoted passages say, quite correctly, that we should take with a grain of salt the idea that a Gödel sentence asserts its own unprovability, this is why. A Gödel sentence is literally just a sentence in the object language of arithmetic. It is only connected to a statement about unprovability via a material biconditional and an encoding of sentences into the language of arithmetic.

Answer 2

Turning David Gudeman's comment into an answer: what it means for a sentence to be self-referential is that it says something about itself, not that it says everything that could be said about itself. It's enough for G_F to assert the unprovability-in-F of G_F for it to be self-referential—dodging the question of whether it really does assert that, which I think is not relevant to the bolded part of your question.

Still dodging that question, G_F definitely doesn't assert its own unprovability in any other formal system than F, or in any broader sense of provability at all. The provability predicate encoded in G_F is specific to F.

Answer 3

The correct conclusion is: The Gödel sentence is either true but has no proof, or it is false and has a proof. A mathematical system powerful enough to express the Gödel sentence is either incomplete (true sentences without proof) or contradictory (false sentences without a proof).

Of course having a contradiction would be quite unfortunate and fatal.