• The idea of a decidable theory • Thm. 3: Any consistent, negation-complete, axiomatized formal theory is decidable • Expressing and capturing properties, relations and functions • The idea of a sufficiently strong theory • Thm. 4: No consistent, sufficiently strong, axiomatized formal theory is decidable • Why the argument for Thm. 4 involves ‘diagonalization’ • Hence, from Thms 3 and 4: A consistent, sufficiently strong, axiomatized formal theory cannot be negation complete...