1 A generalized incompleteness theorem, G¨ 1.1 Notational conventions 1.2 Capturing p.r. functions and relations 1.3 Arithmetizing syntax 1.4 ‘ T is p.r. formalized’ and ‘niceness’ 1.5 ω-consistency, ω-completeness 1.6 The Fixed-point Lemma 1.7 A general version of G¨ odel’s First Theorem 2 Giving the incompleteness theorem more bite 2.1 Confirming that PA is p.r. formalized 2.2 Confirming that PA is p.r. adequate 2.3 Mining the adequacy proof 2.4 Refining the First Theorem 2.5 The ‘syntactic’ vs the ‘semantic’ theorems...