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Introduction to Chapters 3 – 9

If the natural numbers are assumed to be a model of my axioms G then the sentence w , to be exhibited for the incompleteness result (2), will be seman­ tically equivalent to the sentence c saying that it is not the case that the code of w belongs to the codes of provable sentences. While actually a simple consequence of a combinatorial diagonal argument, this situation may appear paradoxical if the fact that L speaks about its arithmetical copy Lg is mistaken to mean that L speaks "about itse lf and that w then would state its own unprovability. Such semantical paradoxes about self-reference, exemplified in that of The Liar, have occupied philosophers since antiquity, and in Chapter 3 I shall discuss this paradox and shall use its analysis to show that, for a specified internal language fragment within an outer langua­ ge, no definition of internal truth can be given, nor can internal provability be complete. All of Chapter 3 can be understood without previous know­ ledge about logic or mathematics, and its results only need that a certain technical hypothesis, connecting the interior with the exterior language, be assumed to hold. Of course, the verification of this hypothesis for concrete situations does require non-trivial mathematical efforts which are the con­ tent of the later Chapters.