TITLE: I don't understand Gödel's incompleteness theorem anymore
QUESTION [52 upvotes]: Here's the picture I have in my head of Model Theory:

a theory is an axiomatic system, so it allows proving some statements that apply to all models consistent with the theory
a model is a particular -- consistent! -- function that assigns every statement to its truth value, it is to be thought of as a "concrete" object, the kind of thing we actually usually think about. It's only when it comes to models that we have the law of the excluded middle.

My understanding of Gödel's first incompleteness theorem is that no theory that satisfies some finiteness condition can uniquely pin down a model.
So I am not really surprised by it. The idea of theories being incomplete -- of not completely pinning down a particular model -- is quite normal. The fact that no theory is complete seems analogous to how no Turing machine can compute every function.
But then I read this thread and there were two claims there in the answers which made no sense to me:

Self-referential statements as examples of unprovable statements -- Like "there is no number whose ASCII representation proves this statement".

A statement like this cannot be constructed in propositional logic. I'm guessing this has to do with the concept of a "language", but why would anyone use a language that permits self-reference?
Wouldn't that be completely defeat the purpose of using classical logic as the system for syntactic implications?
If we permit this as a valid sentence, wouldn't we also have to permit the liar paradox (and then the system would be inconsistent)?

Unprovable statements being "intuitively true/false" -- According to this answer, if we found that the Goldbach conjecture was unprovable, then in particular that means we cannot produce a counter-example, so we'd "intuitively" know that the conjecture is true.

How is this only intuitive? If there exist $\sf PA$-compatible models $M_1$, $M_2$ where Goldbach is true in $M_1$ but not $M_2$, then $\exists n, p, q$ such that $n= p+q$ in $M_1$ but not in $M_2$. But whether $n=p+q$ is decidable from $\sf PA$, so either "$\sf{PA}+\sf{Goldbach}$" or "$\sf{PA}+\lnot\sf{Goldbach}$" must be inconsistent, and Goldbach cannot be unprovable. Right?
In any case, I don't know what it means for the extension to be "intuitively correct". Do we know something about the consistency of each extension or do we not?
Further adding to my confusion, the answer claims that the irrationality of $e+\pi$ is not such a statement, that it can truly be unprovable. I don't see how this can be -- surely the same argument applies; if $e+\pi$'s rationality is unprovable, there does not exist $p/q$ that it equals, thus it is irrational. Right?

REPLY [13 votes]: Self-referential statements as examples of unprovable statements -- Like "[there is no number whose ASCII representation proves this statement][1]".

A statement like this cannot be constructed in propositional logic. I'm guessing this has to do with the concept of a "language", but why would anyone use a language that permits self-reference?

Here's the crux of the issue. Actually, such a statement can be constructed. (Or, at least, a statement which acts like such a statement can be constructed.)
As you know, it's not possible to take the sentence "This sentence cannot be proved in ZFC" and simply translate it directly into the language of ZFC. This is because, as you know, there is nothing in the language of ZFC that means "this sentence".
What we can do, however, is create a sentence G that is true if and only if G cannot be proved in ZFC. How can we do this?
Well, take a look at the following English sentence:

If you write down the following, and then write it again in between quotation marks, then the resulting statement cannot be proved in ZFC: "If you write down the following, and then write it again in between quotation marks, then the resulting statement cannot be proved in ZFC:"

Notice that the part within quotation marks is identical to the part outside of quotation marks, and so "the resulting statement" is identical to the original statement. This statement refers to itself without ever using the phrase "this statement"!
It is possible to do something similar to the above "tricky sentence" in the language of ZFC. The desired sentence is "The sentence with Gödel number $N$ cannot be proved in ZFC", where $N$ is a particular number which is chosen in a way similar to the above "tricky sentence", so that $N$ is the Gödel number for a sentence which is logically equivalent to "The sentence with Gödel number [$N$] cannot be proved in ZFC".
The reason that this can't be extended to form the liar paradox is that the predicate "the statement $p$ cannot be proved in ZFC" can be defined in the language of ZFC, whereas the predicate "the statement $p$ is false" cannot. (In fact, the liar paradox you mention is the proof that the predicate "the statement $p$ is false" cannot be defined in the language of ZFC.)