Reflection principles for the continuum∗

Let HC ′ denote the set of sets of hereditary cardinality less than 2ω. We consider reflection principles for HC ′ in analogy with the Levy reflection principle for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x1, . . . , xn) is a property which is provably persistent in extensions by elements of B, then R(a1, . . . , an) holds whenever a1, . . . , an ∈ HC ′ and R(a1, . . . , an) has a positive IB-value for some IB ∈ B. Suppose C is the class of Cohen algebras. We prove that Con(ZF ) implies Con(ZFC+Max(C)). For a different principle, let CCC be the class of all CCC algebras. We prove that ZF+ Levy schema, and ZFC+Max(CCC) are equiconsistent. Max(CCC) implies MA, while Max(C) implies ¬MA. We give applications of these reflection principles to Löwenheim-Skolem theorems of extensions of first order logic. For example, Max(C) implies that the Löwenheim number of the extension of first order logic by the Härtig quantifier is less than 2ω.