Comment by redvenom | Hacker News Reader

Comment by redvenom | Hacker News Reader

Parent Full thread

The answer you got from lmm and others is wrong (I am a professional mathematician and did research in logic).

The completeness theorem says simply: if T is a first-order theory (list of axioms in first-order logic), any sentence true in every model of T is provable by logical deduction in T.

The first incompleteness theorem says: if T is a consistent recursively enumerable theory that can contains a sufficient amount of Peano arithmetic, then there exists a sentence which is neither true in all models nor false in all models. In other words, there is a sentence which is true in at least one model of T, and false in another.

The second incompleteness theorem says that if T can interpret Peano arithmetic, then we cannot prove the consistency of T within T.

So a tl;dr:

-Completeness: any SENTENCE true in ALL models is PROVABLE - applies to all first order theories

-1st Incompleteness: there EXISTS sentences which are TRUE in some models, FALSE in others. - Applies to theories that contain enough arithmetic

-2nd Incompleteness: if a sufficiently strong system is CONSISTENT, we CANNOT PROVE that CONSISTENCY within the system.

NB: of course, if you have a sufficiently WEAK system, like the axioms of group theory together with "FOR ALL x FOR ALL y (x=y)", then that theory would be COMPLETE and Godel's incompleteness theorem does not apply here.

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.")

Reply on news.ycombinator.com