[FOM] Completeness and Compactness

[Richard Zach]

Infinite-valued Gödel logic is an example, I think. The set of tautologies is intuitionistic logic + (A -> B) \/ (B -> A), so it's weakly complete. But for most infinite truth-value sets, entailment is not compact.

See https://richardzach.org/1998/05/29/compact-propositional-godel-logics/

On 2019-12-06 8:23 p.m., Richard Kimberly Heck wrote:
The recent discussion of axiomatizations of PA has reminded me of a question that's long puzzled me, but for which I think there must be some well-researched answer.

In many proofs of the Gödel completeness theorem, the compactness theorem emerges as not quite as a corollary, but as almost a special case. And, indeed, if one assumes that some logical system S has the property that, (*) whenever Γ ⊩ A (even if Γ is infinite), then Γ ⊢S A, and if one assumes also that S-proofs are finite objects, then compactness follows more or less immediately. I take it that this is what is going on in the first-order case. The proof of completeness establishes (*), and that's it.

So, my question: Are there reasonably natural examples where these come apart? Where there's completeness for finite sets of formulae (which, in many cases, would be the same as completeness for (single) valid formulae), but where compactness fails?
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