The course consists in a detailed exposition of Gödel's First Incompleteness Theorem and its proof. Students will be introduced to first-order axiomatic theories, first-order arithmetic, enumerability, recursivity, definability and representability, models of arithmetic and arithmetical truth, the arithmetical hierarchy, Gödel's coding technique, the Diagonal Lemma, and Gödel sentences. Time permiting, we will also look into Gödel's Second Incompleteness Theorem, Tarski's Theorem, Hibert-Bernays conditions on provability, and Löb's Theorem.