In lieu of an abstract, here is a brief excerpt of the content:

3 On the Shoulders of Giants Our first results concerning axiomatic truth theories are of a negative nature. Gödel proved that no sufficiently strong, consistent mathematical theory proves its own consistency. Using Gödel’s techniques, Tarski’s theorem can be proved. This theorem says that no sufficiently expressive language can define its own truth predicate. Tarski’s theorem on the undefinability of truth in turn has negative consequences for axiomatic theories of truth. It implies that no consistent truth theory implies all the Tarski-biconditionals. 3.1 Introduction Tarski’s theorem on the undefinability of truth is the starting point of contemporary axiomatic theories of truth. Versions of Tarski’s theorem are reviewed in this chapter. This leads us to the realization that “naive” attempts to construct an axiomatic theory of truth fail miserably. We also review other metatheorems that are put to use in the chapters to come: Gödel’s two incompleteness theorems and Gödel’s completeness theorem. In addition, it is indicated how in the language of Peano arithmetic truth definitions for arithmetical sentences of bounded complexity can be articulated. I do not present all details of the proofs in this chapter. For the details, the reader is referred to the literature. A good source for many of these proofs is [Goldstern & Judah 1998, chapter 4]. There exist weaker and stronger versions of both Gödel’s two incompleteness theorems and Tarski’s undefinability theorem. Detailed proofs of the weaker versions of these metatheorems are really not hard to give. So it is misleading to speak of the incompleteness theorem and of the undefinability theorem: Several incompleteness and undefinability theorems can be distinguished. It is better to speak, in the spirit of Goldstern and Judah, of the incompleteness phenomenon and of the undefinability phenomenon. 28 Chapter 3 Once one has gone through the proofs of the weaker versions of the metatheorems , it becomes plausible that the stronger versions also hold. But if one insists on the details of these stronger versions, then more (somewhat tedious) work needs to be done. Those who do want to walk the extra mile are referred to [Boolos & Jeffrey 1989, chapters 15 and 16]. 3.2 Coding in the Language of Peano Arithmetic For a while we are sticklers for notation. This is just to refresh the reader’s mind about how it is done according to the rules of the art. Then from chapter 4 onward, in the interest of readability, we abandon all notational scruples and express ourselves in a carefree, non-well-formed manner. But by then it should be clear how our non-well-formed stammering can be transformed into coherent speech. We first revisit the first-order language LPA of Peano arithmetic. The logical vocabulary of LPA contains the propositional operators of conjunction (∧) and negation (¬). Disjunction, material implication, and material equivalence are defined (in the metalanguage) in terms of ¬ and ∧ in the usual manner. Furthermore, LPA contains the universal quantifier ∀. The existential quantifier is again defined in terms of ∀ and ¬ in the usual manner. Furthermore , LPA contains an infinite stock of variables xi for i ∈ N and the identity relation =. The mathematical vocabulary of LPA consists of a successor function symbol s, a zero constant 0, and function symbols + and × for addition and multiplication. Using the successor function symbol and the zero constant, we can build standard numerals for natural numbers. For each number n ∈ N, the standard numeral for n consists of the zero constant prefixed by n copies of the successor symbol. The standard numeral for n is denoted as n. Next, the elements of the vocabulary of LPA are algorithmically assigned unique numerical codes (gödel codes or gödel numbers) according to some coding scheme. We need not go into the details of this coding scheme here, but, for definiteness, let us say that we adopt the coding scheme of [Goldstern and Judah 1998, p. 209]. Then, given some further conventions, this coding scheme can be extended to terms, formulae, sentences of LPA. If e is a simple or complex expression of LPA, then we denote its gödel number as e.1 1. The idea of coding the structure of a language using natural numbers can be traced back to Leibniz. But Gödel was the first to work out accurately and in detail how this project should be executed. [3.16.81...

Share