Seq-consistency property and interpolation theorems
In infinitary logics with conjunctions over non denumerable sets of formulas and quantification over non finite sets of individual variables Craig's interpolation theorem does not hold. Changing the notion of satisfiability to that of \(\omega\)-satisfiability Karp was able to extend Craig's interpolation to the infinitary languages with conjunctions and quantifications over sets of cardinality a strong limit cardinal of cofinality \(\omega\). But in Karp's interpolation theorem, whereas it is required that an implication be \(\omega\)-valid, the interpolant is a formula with the usual limitation on the symbol such that it simply is implied (not \(\omega\)-implied) by the antecedent and simply implies (not \(\omega\)-implies) the consequent. To prove this result Karp used the tool of the consistency properties, but her notion of a consistency property was such that if a set of sentences belongs to the consistency property then it is \(\omega\)-satisfiable, but not always an \(\omega\)-satisfiable set of sentences belongs to some consistency property. Cunningham improved Karp's interpolation theorem as to have the \(\omega\)-implication in the definition of the interpolant, but she didn't touch the notion of consistency property. In the paper under consideration, instead of sets of sentences, \(\omega\)-sequences of sets of sentences are considered, and the notion of \(\omega\)-satisfiability is extended to such objects. Then the notion of seq-consistency property is introduced and it is proved that if an \(\omega\)-sequence of sets of sentences belongs to a seq- consistency property then it is \(\omega\)-satisfiable. It is also proved that if an \(\omega\)-sequence of sets of sentences is \(\omega\)-satisfiable then it belongs to a consistency property. Finally Cunningham's interpolation theorem is reobtained using this new tool. In the conclusion it is suggested to try the new technique to tackle other interpolation theorems.
scientific article; zbMATH DE number 3847368