Intuition behind Gödel's diagonal lemma?

Question

I'm trying to develop a good intuition for how Gödel's first incompleteness theorem works. I think I've got most of it, but there's one step involving a lemma I haven't been able to really get.

Here's my sense of how the proof works:

1. Show there's a way to give each formula and proof a unique natural number (its "Gödel encoding").
2. Show that this lets formulas refer to formulas and proofs. Intuitively this lets us write formulas that say things like "The formula with Gödel encoding $x$ has proof $P$."
3. Construct a variable formula $f(x)$ that in effect says "The formula with Gödel encoding $x$ has no proof."
4. Via a magical "diagonal lemma", we know there's a natural number $n$ such that the Gödel encoding of $f(n)$ is $n$.
5. $f(n)$ is, in effect, the statement "This statement is unprovable." If it's true, then it's unprovable, meaning there are true statements the system cannot prove — i.e. it's incomplete. If $f(n)$ is false, then it's provable, meaning the system can prove false things and is inconsistent. QED.

I'd like some kind of intuition for that diagonal lemma in step 4 that feels as intuitive to me as the rest of the argument. LLMs haven't been helpful here; they keep either walking through their version of the proof, or saying extremely vague things like "The lemma involves self-reference."

Another way to say it is, I'm looking for the "key" that "unlocks" the diagonal lemma. Roughly the way that the statement "This statement is unprovable" sort of "unlocks" the overall proof: if I know that's what I'm aiming for, and why, then the formalism is mostly about checking the details of how I construct that statement.

As it is, I can't intuitively feel why the diagonal lemma wouldn't apply to the formula that says in effect "The formula with Gödel number $x$ is false." That'd let us construct the statement "This statement is false", which has a paradoxical truth value. I'm sure the lemma doesn't allow this, but I'm deducing this socially, not mathematically. (Maybe we can't construct formulas that talk about other formulas' truth values? If so, I'm not sure why not.)

I know there are a few other discussions of this already:

• Here is someone asking about the details of the diagonal lemma. I think I understand the answer to their questions. That doesn't give me a feel for why the lemma is true though.
• Then there's this one, which goes into way too much detail to give me the intuitive overview I'm looking for, and is about a possible alternative way of proving the lemma.
• Someone tagged my question as duplicating this one. I like the intuitive style the answerer gives; it's exactly the right type I'm looking for. And he explains what the lemma needs to show in good clear terms. That's super a step in the direction I'm hoping for. But his explanation for the proof of the lemma seems to me to be almost entirely contained within the statement "it turns out that". ("It turns out that if you calculate the number of this formula, you get exactly the number you get by starting with the number $k$ and performing the operations described by the statement.")

I've stared at various descriptions of the lemma's proof, including its proof in Wikipedia. It feels like pressing my brain through a symbolic meat-grinder. If I stare very, very closely at each step, I can agree it follows from previous steps. But I have no idea what intuition guided anyone to write those lines. I'm way more interested in the intuition than I am in the minute details of rigor here: I can make sense of the latter with the former, but I haven't been able to make it work the other way around.

Do any of you have a guess about a "key" that might help me make sense of the diagonal lemma?

Answer 1

Godel numbering immediately enables "self-plugging-in:" given a formula $\psi(x)$ with Godel number $n$, let $\psi^*$ be the sentence $\psi(\underline{n})$ (where $\underline{n}$ is the numeral corresponding to $n$). This is completely straightforward.

Now this doesn't yet result in self-reference since $\psi^*$ just "refers to" $\psi$. On the other hand, since $\psi^*$ was built from $\psi$, it feels like we're pretty close: we have a sentence which refers to the seed of itself. We just need to catch our tail so to speak.

To do this, we abstract a bit: the function sending the Godel number of a formula to the Godel number of that formula's starred-version is itself nicely definable (namely: representable). So for any property $P$ of sentences that we can encode in the language of arithmetic, we can use this function to write a formula $\sigma_P(x)$ which intuitively says "The sentence $\chi^*$ (where $\chi$ is the formula with Godel number $x$) has property $P$."

• EDIT: note that the vicissitudes of first-order logic's syntax make this last step rather messy in a pretty stupid way. We have a (definition of a) function $f$ sending the Godel number of $\chi$ to the Godel number of $\chi^*$, and we want $\sigma_P(x)$ to just be $P(f(x))$. But that's not how FOL works: since the definition of the function $f$ is itself a formula $\varphi_f$, we have to instead phrase $\sigma_P(x)$ as $$\forall y[\varphi_f(x,y)\implies P(y)]$$ (or $\exists y[\varphi_f(x,y)\wedge P(y)]$ if you prefer). Really this is a historical artifact: if computer science had developed before first-order logic, we would (I hope!) have standardized a more straightforward notation for working with defined functions.

I claim that this lets us catch our tail as desired. Let $k$ be the Godel number of $\sigma_P$. Then $\sigma_P(\underline{k})$ says roughly "If you plug $\sigma_P$ into itself you get a sentence with property $P$." But $\sigma_P(\underline{k})$ is exactly the sentence you get if you "plug $\sigma_P$ into itself." We can now visualize this nicely in natural language (I think this particular phrasing is due to Hofstadter):

"Yields a sentence with property $P$ when preceded by its quotation" yields a sentence with property $P$ when preceded by its quotation.

Answer 2

Edit:

As it is, I can't intuitively feel why the diagonal lemma wouldn't apply to the formula that says in effect "The formula with Gödel number $x$ is false." That'd let us construct the statement "This statement is false", which has a paradoxical truth value. I'm sure the lemma doesn't allow this, but I'm deducing this socially, not mathematically. (Maybe we can't construct formulas that talk about other formulas' truth values? If so, I'm not sure why not.)

This is a great question and it has a great answer. The answer is that this argument you've just given is a proof by contradiction that it is not possible to say "this statement is false" at all! Formally, it is not possible to construct a truth predicate in, say, Peano arithmetic, meaning a predicate $\text{True}(\ulcorner \varphi \urcorner)$ which is true exactly for the Godel codes of true statements. This is Tarski's undefinability theorem.

This says in some sense that truth, unlike provability, is not "finite." We really do not have a better way to define truth than to say, for example, that $\forall x P(x)$ is true precisely when $P(x)$ is true of all ("actual") natural numbers $x$, which is in some sense a fundamentally infinitary statement.

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Personally I've never understood the diagonal lemma either. There's this intriguing paper by Yanofsky, A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points, which claims that all diagonal arguments are essentially the same: the diagonal lemma, Cantor's theorem, Russell's paradox, etc. Specifically it claims that they are all essentially the same as the Lawvere fixed point theorem; this observation is due to Lawvere. I think this is quite a beautiful idea, but I've never understood this paper either! But perhaps the real key is lurking in that paper.

Fortunately, there are ways to prove the incompleteness theorems without using the diagonal lemma. The proof that makes the most sense to me is explained by Scott Aaronson here, and he introduces it as follows:

First, though, let's see how the Incompleteness Theorem is proved. People always say, "the proof of the Incompleteness Theorem was a technical tour de force, it took 30 pages, it requires an elaborate construction involving prime numbers," etc. Unbelievably, 80 years after Gödel, that's still how the proof is presented in math classes!

Alright, should I let you in on a secret? The proof of the Incompleteness Theorem is about two lines. The caveat is that, to give the two-line proof, you first need the concept of a computer.

The proof is by reduction to the halting problem. The self-reference in this argument comes from the familiar halting problem arguments where you feed a program its own source code and so forth. Once you have that:

As I said, once you have Turing's results, Gödel's results fall out for free as a bonus. Why? Well, suppose the Incompleteness Theorem was false -- that is, there existed a consistent, computable proof system F from which any statement about integers could be either proved or disproved. Then given a computer program, we could simply search through every possible proof in F, until we found either a proof that the program halts or a proof that it doesn't halt. (This is possible because the statement that a particular computer program halts is ultimately just a statement about integers.) But this would give us an algorithm to solve the halting problem, which we already know is impossible. Therefore F can't exist.

Other approaches to the incompleteness theorem are also possible, which are "powered by" other paradoxes; if we think of the diagonal lemma proof as being "powered by" the liar paradox, there's also a proof which is "powered by" the Berry paradox. This idea appears to be due to Vopěnka and Chaitin and a detailed version of the proof is given in Maroš Grego's The incompleteness theorems and Berry’s paradox.