Completeness and Applications
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The other way around—true ⇒ derivable—is studied in a separate chapter. Here Gödel’s completeness theorem: Truth = derivability, is proved following the method of Leon Henkin. Here a bit of set theory comes in, e.g. classes of structures, extensions, etc. The student need not worry, all prerequisites are lined up—consistent, maximally consistent, conservative extension, witness. The key lemma, the model existence lemma, turns out to be a gentle piecing together of structures. The techniques are used to prove a number of results that take us to model theory, such as the compactness theorem: if all finite parts of a theory are consistent, so is the theory itself. The Skolem-Löwenheim theorems provide (for most structures) larger or smaller structures that are logically indistinguishable from the original one (elementary equivalence, elementary extension). Questions like: Is a particular class of structures (say, algebraically close fields) axiomatizable? Which theories are decidable? are handled by model theoretic means. Skolem considered the possibility of introducing a function picking suitable elements in a structure, once ∀x∃yφ(x,y) has been shown. The basic facts of these Skolem functions are discussed. The famous Herbrand theorem is found in the exercise section. The chapter ends with a new section on ultraproducts, i.e. structures produced by a clever product construction from given structures. An example is a non-standard extension of the natural number system. The topic is on the miraculous side of our logic course, as it shows us that it is perfectly possible to obtain logical results in a logic-free way.