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Mathematical Logic I (H,Q) is the first semester of a year long course studying the logical methods used in mathematical reasoning. The first semester explores the construction of formal languages in which to cast mathematical discourse, introduces systems of proof for deriving propositions from assumptions, and develops a formal semantics that provides a precise criterion of logical consequence. We expect a system of proof to allow the derivation only of propositions that are logical consequences of the assumptions (soundness). A principal result establishes the converse: these systems of proof are such that any logical consequence is derivable (completeness). This provides us with a purely mathematical characterization of logic within which mathematical theories can be formulated and their properties studied (decidability, axiomatizability, consistency, completeness), a pursuit commonly known as metamathematics.