Abstract
We first raise an issue on a view that Gödel’s completeness theorem assures that first-order logic gives an adequate formalization of the concept of proofs in ordinary mathematics. Then we discuss philosophical backgrounds of Henkin’s proof of the completeness theorem and their effects on the change of interpretations of first-order and second-order quantifiers. Lastly, we analyze the meanings of formal consistency statements and their relation to Hilbert’s program, and we claim that the relationship between the concepts of formal and informal proofs is closely connected to interpretations of quantifiers.
