50 Logic as a Tool Meanwhile, in the early 1930s some groundbreaking results in logic were announced by a young logician having just completed his doctoral studies, Kurt Gödel Those results, that made Gödel the most famous logician of the 20th century, showed that Hilbert’s idea was only partly realizable, in a sense that purely logical validity and consequence in first-order logic can be axiomatized completely by Gödel’s completeness theorem for first-order logic; it was not, however, realizable for the richer notion of mathematical consequence, even in the relatively simple mathematical systems of arithmetic of natural numbers with addition and multiplication More precisely, in 1931 he proved his two celebrated Gödel’s incompleteness theorems The first of these stated that no sufficiently expressive and reasonably axiomatized (with an effectively recognizable notion of axioms) deductive system, such as Peano’s system of arithmetic, can be complete The second incompleteness theorem claimed that such a theory cannot even prove its own consistency – suitably encoded as a formula in that system – unless it is inconsistent (in which case it can derive any formula by using the sound logical rule Ex Falso Quodlibet) No absolute and finitary proof of consistency of such system is therefore possible Hilbert’s dream of formalizing the whole of mathematics in a provably consistent way turned out to be unattainable Still, deductive systems and their proof theory have remained one of the main directions for development of modern logic 2.1.3 Soundness, completeness and adequacy of deductive systems A very important aspect of deductive systems is that derivations in them are completely mechanizable procedures that in principle not require any intelligence or understanding of the meaning of the formulae or rules involved; in fact, such a meaning need not be specified at all Derivations in a given deductive system can therefore be performed by a mechanical device such as a computer without any human intervention, as long as the axioms and rules of inference of the deductive system have been programmed into it While deductive systems are not explicitly concerned with the meaning (semantics) of the formulae they derive, they are designed with the purpose of deriving only valid logical consequences from the assumptions A deductive system with this property is called sound In particular, every theorem of a sound deductive system must be a valid formula, that is, in the case of propositional logic it must be a tautology Formally, a deductive system D is sound (or correct) for a given logical semantics (that is, well-defined notions of logical validity and consequence) if D can only derive logically valid consequences, that is: A1 , , An D C implies A1 , , An |= C In particular: D C implies |= C A deductive system D is complete for a given logical semantics if D can derive every valid logical consequence (as defined in that semantics), that is: A1 , , An |= C implies A1 , , An In particular, |= C implies logically valid formula D D C C , that is, a complete deductive system can derive every