What Gödel Discovered

It's quite a fun effect! Hofstadter's claim that "I" (our conciousness) is an epiphenomena was very eye opening for me.

Yes if you appeal to model theory you're on firm ground with truth. But that's not how most informal introductions to the incompleteness theorems go (and indeed not how this article went about it).

More to the point, Godel's incompleteness theorems are theorems of proof theory, not model theory. Their proofs can be carried out with no recourse to models at all. And I think they are far less prone to being misunderstood when carried out without recourse to models and by extension without recourse to semantic truth.

You can of course choose to interpret them in model theory if you're working in a semantically complete logical system such as first-order logic with the usual semantics or second-order logic under Henkin semantics, and it is often illuminating to do so once you have those fundamentals. But I think in an informal setting those will tend only to confuse rather than clarify. And importantly the incompleteness theorems still hold in contexts where you don't really have a meaningful model theory to speak of.

Very interesting. I'm wondering if this has implications in physics ? Since "the book of nature is written in mathematical language", does this mean that attempts to find a unified theory of everything are doomed ? I'm thinking not necessarily, since we wouldn't need to prove all possible physical phenomena, only those that actually happen, but how do we know if they are among the one that can be proven ?

Ah... this also has nuances. Although perhaps the unspoken assumption of your post is that you are trying to explain the incompleteness theorems through the lens of first-order logic? In which case your definitions are completely fine.

But just to be clear, your summarization of the first incompleteness theorem is through the lens of model theory and therefore predicated on the completeness theorem and doesn't hold in its absence. The canonical example is second-order PA under full second-order semantics, which has only one model (this is also true of second-order ZFC). Hence there do not exist sentences of second-order PA which are true in some models and false in others.

To fully flesh out the syntactic results of Godel's incompleteness theorems and their semantic consequences when Godel's completeness theorem holds, there's the following syntactic -> semantic chains.

1st incompleteness: there exist sentences which cannot be proved within any theory containing enough arithmetic. Therefore by Godel's completeness theorem there are sentences in the theory which are satisfied by some models and whose negation are satisfied by other models.

2nd incompleteness: the sentence Con(T) corresponding to the informal sentence "the theory T is consistent" cannot be proved within the theory T itself assuming T contains enough arithmetic. Therefore by Godel's completeness theorem there are models of T where Con(T) is satisfied and models of T where Not(Con(T)) is satisfied.

As I said in another post while Godel's completeness theorem is an essentially model-theoretic result (and it is the foundation that all other model theory builds on), Godel's incompleteness theorems are fundamentally syntactic results that live in the realm of proof theory, rather than semantic ones that live in the realm of model theory, although they have deep implications for the latter.

(Also to be extraordinarily pedantic, as you point out elsewhere, a sentence is true if it is satisfied by all models. Hence technically it is a misnomer to say that a sentence is true in a given model. Rather a sentence is satisfied by a given model. But it's a common enough bit of informal terminology, but I just want to point it out for people who aren't familiar with model theory so that they don't get tripped up trying to square different formal definitions of "true.")

Oi, thanks for the kind words and the catch! Updated (should take a few minutes to show up)