chapter  4
28 Pages

Undefinability and Incompleteness, General Theory

In this Chapter I repeat the approach of Chapter 3, taking as exterior a language from mathematical logic, and as interior its arithmetical mirror obtained with help of coding functions for terms, formulas and derivations; the basic technical hypothesis now becomes the representability of certain arithmetical relations, mirroring syntactical relationships under the coding. This notion is explained in Section 4.2; the following three Sections first present TARSKl’ s 36 theorem on the undefinability of truth, and then in­ completeness theorems of the type of GODEL’ s 31 and ROSSER’ s 36 , diffe­ rentiated in particular by their underlying semantical assumptions. Common to the proofs of these theorems is the use of fixpoint arguments as develo­ ped by Tarski in TARSK1-M0ST0WSKI-R0B1NS0N 51, and a simple combi­ natorial frame based on this idea is presented in Section 4.1.