Abstract
The abstract Lindenbaum lemma is a crucial result in algebraic logic saying that the prime theories form a basis of the closure systems of all theories of an arbitrary given logic. Its usual formulation is however limited to nitary logics, i.e., logics with Hilbert-style axiomatization using nitary rules only.
In this contribution, we extend its scope to all logics with a countable axiomatization and a well-behaved disjunction connective.We also relate Lindenbaum lemma to the Pair extension lemma, other well-known result with many applications mainly in the theory of non-classical modal logics. While a restricted form of this lemma (to pairs with nite right-hand side) is, in our context, equivalent to Lindenbaum lemma, we show a perhaps surprising result that in full strength it holds for nitary logics only.
Finally we provide examples demonstrating both limitations and applications of our results.