[FOM] Completeness and Compactness

[Richard Kimberly Heck]

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?

Riki



-- 
----------------------------
Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

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