 In addition, mathematics contains machinery for a metalanguage, that is for talking about mathematics itself. This includes the concepts: (i) names for mathematical statements and other parts of syntax, (ii) self-reference, (iii) proof and (iv) truth. Gödel’s contribution to the philosophy of mathematics was to show that the first three of these can be rigorously expressed in arithmetical theories, albeit in theories which are either inconsistent or incomplete. The possibility of a well-structured example of the former of these two alternatives, namely inconsistency, was not taken seriously, again because of belief in ECQ. However, in addition natural languages seem to have their own truth predicate. Combined with self-reference this produces the Liar paradox, “This sentence is false”, an inconsistency. Priest (1987) and Priest, Routley and Norman (1989, p. 154) argued that the Liar had to be regarded as a statement both true and false, a true contradiction. This represents another argument for studying inconsistent theories, namely the claim that some contradictions are true, also known as dialetheism. Kripke (1975) proposed instead to model a truth predicate differently, in a consistent incomplete theory. We see below that incompleteness and inconsistency are closely related.