ABSTRACT
IX.1 Gödel spoke of his argument being “closely related to the ‘Liar’” (1931, p. 149), and he meant the Eubulidean Liar, the antinomy in which a man says that he is lying or that what he says is not true, so apparently providing a proposition that says about itself that it is not true. However that may be (cf. VII.2, esp. fn. 1), there is undoubtedly a close relation between it and the Epimenidean Liar, the antinomy in which he only says something to the effect that what he says is not true-something, say, that p, that has the property of being true just in case that it is not true that p also has that property (though saying, as St Paul quotes Epimenides the Cretan as saying, that the Cretans are always liars only half-fulfils this condition: cf. Prior 1958, p. 261). For, taking Tr to be a representation of
‘n.n is the Gödel number of a form that is standardly assigned ’, let L be Carnap’s fixed-point for then L has the property of being standardly assigned just in case also has that property. So now we have relatives of the four components of Gödel’s argument for Theorem VIII.1.1:
1-for any form X, if X is standardly assigned so is 2-if is standardly assigned so is ~L; 3-if is standardly assigned so is L; and 4-for any form X, if X is not standardly assigned then is
standardly assigned . These four propositions are incompatible-which establishes the
falsity of the assumption that the predicate mentioned is representable: THEOREM IX.1.1. TARSKI’S THEOREM. There is no representation of
‘n.n is the Gödel number of a form that is standardly assigned ’.
