TITLE: What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
QUESTION [38 upvotes]: I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by questions two previously asked questions:
(Proofs of Gödel's theorem)
(When are two proofs of the same theorem really different proofs)
To give an example of what I mean: The Godel/Rosser proof (see http://www.jstor.org/pss/2269059 for an exposition) shows that any consistent sufficiently strong axiomatizable theory is incomplete. The proof uses a substantial amount of recursion theory: the representability of primitive recursive functions and the diagonal lemma (roughly the same as Kleene's Recursion Theorem) are essential ingredients. The second incompleteness theorem - that no consistent sufficiently strong axiomatizable theory can prove its own consistency - is essentially a corollary to this proof, and a rather natural one at that. On the other hand, in 2000 Hilary Putnam published (https://doi.org/10.1305/ndjfl/1027953483) an alternate proof of Godel's first incompleteness theorem, due to Saul Kripke around 1984. This proof uses much less recursion theory, instead relying on some elementary model theory of nonstandard models of arithmetic. The theorem proven is slightly weaker, since Kripke's proof requires $\Sigma^0_2$-soundness, which is stronger than mere consistency (although still weaker than Godel's original assumption of $\omega$-consistency).
Kripke's proof is clearly sufficiently different from the Godel/Rosser proof that it deserves to be considered a genuinely separate object. What makes the difference seem really impressive, at least to me, is that Kripke's proof yields a different corollary than that of Godel/Rosser. In a short paragraph, Putnam shows (and I do not know whether this part of his paper is due to Kripke) that Kripke's argument proves that there is no consistent finitely axiomatizable extension of $PA$. This is not a result which I know to follow from the Godel/Rosser proof; moreover, the Second Incompleteness Theorem, which is a corollary to Godel/Rosser's proof, does not seem easily derivable from Kripke's proof.
Motivated by this, say that two proofs of (roughly) the same theorem are essentially different if they yield different natural corollaries. Clearly this is a totally subjective notion, but I think it has enough shared meaning to be worthwhile.
My main question, then, is:

What other essentially different proofs of something resembling Godel's First Incompleteness Theorem are known? In other words, is there some other proof of something close to "every consistent axiomatizable extension of $PA$ is incomplete" which does not yield Godel's Second Incompleteness Theorem as a natural corollary?

I am especially interested in proofs which don't yield the nonexistence of consistent finitely axiomatizable extensions of $PA$, either, and in proofs which do yield some natural corollary. I don't particularly care about the precise version of the First Incompleteness Theorem proved: if it applies to systems in the language of second-order arithmetic, if it assumes $\omega$-consistency, or if it only applies to systems stronger than $ATR_0$, say, that's all the same to me. However, I do require that the version of the incompleteness theorem proved apply to all sufficiently strong systems with whatever consistency property is needed; so, for example, I would not consider the work of Paris and Harrington to be a good example of this.
The only other potential example of such an essentially different proof that I know of is the proof(s) by Jech and Woodin (see https://andrescaicedo.files.wordpress.com/2010/11/2ndincompleteness1.pdf), but I don't understand that proof at a level such that I would be comfortable saying that it is in fact an essentially different proof. It seems to me to be rather similar to the original proof. Perhaps someone can enlighten me?
Of course, entirely separate from my main question, my characterization of the difference between the specific two proofs of the incompleteness theorem mentioned above may be incorrect. So I'm also interested in the following question:

Is it in fact the case that Kripke's proof does not yield Second Incompleteness as an natural corollary, and that Godel/Rosser's proof does not easily yield the nonexistence of a consistent finitely axiomatizable extension of PA as a natural corollary?

REPLY [2 votes]: There’s also an “invariant definability” argument. I’ll sketch it quickly below, and then give an analysis to explain why I think it’s meaningfully different. Embarrassingly I can't find a source for it at the moment; I recall seeing it as a footnote in Kreisel's Model-theoretic invariants paper, but it doesn't seem to be there. Multiple authors have written on invariant definability (which this answer is not-so-secretly an advertisement of) so I haven't yet been able to conduct an exhaustive search for the reference, but when I find it I'll update this. Incidentally, this argument was referred to at the beginning of another answer of mine.
Below, “definable” means “definable without parameters.” For more pleasant language I'll call this argument the "Tarskian argument."

Argument
Let $T$ be an “appropriate” theory of arithmetic (say, $T\supseteq R$). We tweak Tarski’s undefinability theorem very slightly as follows. For $X\subseteq\mathbb{N}$ and $M\models T$, say that $X$ is
pseudo-definable in $M$ if $X=D\cap \mathbb{N}$ for some definable $D\subseteq M$. We then have:

$(*)\quad$ Suppose $M\models T$. Then $Th(M)$ is not pseudo-definable in $M$.

The standard proof still works: supposing to the contrary that $\theta$ pseudo-defined $Th(M)$ in $M$, let $m$ be the Godel number of the formula $\eta(x)\equiv$ "$\theta$ fails on the number of the sentence gotten by plugging $x$ into the formula with number $x$" - appropriately formalized - and consider the sentence $\eta(\underline{m})$ (applying representability appropriately).
With this in hand we argue as follows. Suppose $S\supseteq T$ is computable and satisfiable. Then by representability we have that $S$ is pseudo-definable in $M$ for every $M\models T$. Taking $M\models S$, we have by $(*)$ that $S\not=Th(M)$, so $S$ is not complete.

Analysis
Now let me argue in favor of the Tarskian argument being a genuine variation.
First, there’s an easy negative observation: it applies to Willard’s self-verifying theories and so cannot yield the second incompleteness theorem as a direct corollary. More subjectively, the argument is fairly non-constructive, and doesn’t (as far as I can see) quickly yield a specific undecidable sentence.
Of course, this is also a feature of the many standard computability-theoretic arguments. I think there’s still a difference here - this time a positive one - due to the way the Tarskian argument interacts with the notion of invariant definability. There are a couple ways to frame this - see e.g. the beginning of the article Abstract Computability and Invariant Definability by Moschovakis for some discussion - and I'll use the following:

Definition:

An arithmetic context is a set $\mathfrak{C}$ of models of Robinson's arithmetic $R$.

For an arithmetic context $\mathfrak{C}$, a set $A\subseteq \mathbb{N}$ is $\mathfrak{C}$-invariantly definable if there is some formula $\varphi$ with $\varphi^M\cap\mathbb{N}=A$ for all $M\in\mathfrak{C}$.



(Here I’m indulging in the usual abusive conflation of $\underline{k}^M$ and $k$.)


For a theory $E$ and an arithmetic context $\mathfrak{C}$, say that $E$ is $\mathfrak{C}$-satisfiable if some member of $\mathfrak{C}$ satisfies $E$.


Then the Tarskian argument in fact gives:

Proposition: Suppose $\mathfrak{C}$ is an arithmetic context. Then no $\mathfrak{C}$-satisfiable theory is $\mathfrak{C}$-invariantly definable.

(Since the computable sets are invariantly definable in every arithmetic context and every theory extending $R$ yields an arithmetic context, this is a generalization of essential undecidability.)
The point is that in general the $\mathfrak{C}$-invariantly definable sets need not support a good computability theory in any sense: by judiciously terrible choice of $\mathfrak{C}$, we can make the set of $\mathfrak{C}$-invariantly definable sets have basically no structural properties besides forming a Boolean algebra. So, for example, I don’t see how to whip up an analogue of the “inseparable c.e. sets” argument for arbitrary arithmetic contexts.
Of course, pathological arithmetic contexts are uninteresting, so it’s hard to argue that this aspect is actually valuable in any way. But it is - as far as I can tell - a nontrivial feature.