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From my limited reading, is it correct that model theory is the same as theory about semantics of logical systems? Why separate model theory from logic? Thanks.

Tim
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    Proof-theoretic semantics is a research field/philosophy based on the idea that a semantics for logic can be given not only in terms of models, but proofs; that is, the meaning of a connective is defined by e.g. its elimination and introduction rules in a natural deduction proof system. Model-theoretic semantics is one way of doing semantics for logic, but it is not the only semantics there is; not everyone would agree with the distinction "model theory = semantics" and "proof theory = syntax". – Natalie Clarius Sep 01 '20 at 09:25
  • @lemontree That's a great point, and something I omitted in my answer. – Noah Schweber Sep 01 '20 at 14:29
  • @lemontree "Model-theoretic semantics is one way of doing semantics for logic, but it is not the only semantics there is." Do you mean proof system is another semantics for logic? – Tim Sep 06 '20 at 17:11
  • @Tim Yes. The SEP article I linked is a good place to start in case you're interested. – Natalie Clarius Sep 06 '20 at 18:08

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Model theory and logic aren't separated; model theory is a subfield of logic, the other three main subfields being proof theory, set theory, and computability theory. Keep in mind that taxonomies and delineations of mathematical subfields are entirely sociological, so there's a lot of subjectivity here.

The slogan "model theory is semantics and proof theory is syntax" is pretty good but not quite accurate in my opinion. Rather, I'd say that model theory studies the interplay between semantics and syntax (consider results like "every sentence preserved under taking substructures is equivalent to a $\forall$-sentence," which are model-theoretic yet refer to syntax directly) while proof theory studies the syntactic side of things in a self-contained way (that is, proofs and theories as mathematical objects in their own right). It is certainly true that model theory has purely semantic aspects, but it doesn't ignore syntax entirely. (The most "syntax-free" we get is with abstract model theory, but that's really a niche subject - model theory of first-order logic and its relatives is very much syntax-aware.)

Computability theory and set theory then are the more "accidental" aspects of logic - they don't really amount to the study of logic itself, but rather we've singled out two topics in mathematics (computations and sets) as having a particularly logical nature.

I'd say that the unifying thread justifying the collection of these four topics under the single heading of "logic" is that they do a decent job of capturing a particularly philosophical aspect of mathematics. Set theory amounts to the ontology - rejecting formalism at least on a tentative basis, what exactly are mathematical objects and how do we relate them to each other? (Note that this means that I'd fold other foundational theories, like homotopy type theory, into the same "bubble.") Model theory and computability theory capture different aspects of epistemology - how do we use language to describe mathematical objects, and how do we distinguish between different types of (idealized) mathematical knowledge? Finally, proof theory provides an alternative to set theory: we can reject ontology altogether (mostly anyways) and just look at the self-contained apparatus of mathematical language on its own.

Of course one may argue that that's a totally post hoc justification of a contingent state of affairs, and I'll admit that's at least partially true. Nevertheless, I have come around to the stance that there is a real unifying thread here which makes sense somehow.

(Less charitably, these four areas are more connected to each other than to anything else - model theory being the closest to an outlier, given its connections to algebra and some analytic topics, but still suffering from some major isolation.)

Noah Schweber
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  • Re "Computability ... don't really amount to the study of logic", how would you characterize the Curry-Howard isomorphism (e.g., https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence) with respect to that statement? Maybe it amounts to the study of constructive/intuitionistic logic? Or what? – John Forkosh Sep 01 '20 at 04:39
  • @JohnForkosh I should have been more careful in my phrasing (I have to go to sleep but I'll edit tomorrow). What I meant was that computability and set theory aren't a priori about the fundamental apparatus of logic in the sense of truth and entailment. Curry-Howard draws a connection, but it's not built into the definition of the topic. Things like Curry-Howard are evidence for the reasonableness of grouping computability with model/proof theory, but they happen "after the fact," so to speak. – Noah Schweber Sep 01 '20 at 05:08
  • @JohnForkosh Put another way, I can imagine computability theory developing on its own without first-order logic being even considered, as simply the "idealized" abstract theory behind physical computers (and its various generalizations, like oracle machines and Turing degrees). You don't need to talk about sentences and structures to ask whether there are noncomputable incomplete c.e. sets. – Noah Schweber Sep 01 '20 at 05:38
  • Thanks. https://math.stackexchange.com/questions/3806298/is-models-of-the-first-order-logic-system-defined-in-terms-of-some-first-ord is about the dependency between logic and set theory. – Tim Sep 01 '20 at 15:00
  • Thanks again. I think by "proof theory", you meant https://en.wikipedia.org/wiki/Proof_calculus. I notice the similarity and difference between the components of a proof system and a logical system (when S is given) – Tim Sep 06 '20 at 17:13
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    @Tim No, I mean what I wrote (although of course proof calculi are a/the fundamental topic in proof theory). – Noah Schweber Sep 06 '20 at 18:00