Accepting inconsistencies from the paradoxes

It has been proposed that the law of non-contradiction be revised to permit the simultaneous truth and falsity of the key sentences of the logical paradoxes, e.g., This sentence is false. In an attempt to show to what extent this bizarre suggestion of inconsistent models or truth-value gluts is a coherent suggestion it is proved that a first-order language for number theory can be semantically closed by having its own global truth predicate under some non-standard interpretation and thus that it actually can contain the Liar sentence. It is proved that in this interpretation the Liar sentence is both true and false, although not every sentence is.