Lindenbaum and pair extension lemma in infinitary logics

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 finitary logics, i.e., logics with Hilbert-style axiomatization using finitary 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 finite right-hand side) is, in our context, equivalent to Lindenbaum lemma, we show a perhaps surprising result that in full strength it holds for finitary logics only. Finally we provide examples demonstrating both limitations and applications of our results.