220 Logic as a Tool Kurt Friedrich Gödel (28.4.1906–14.1.1978) was an Austrian–American logician, mathematician, and philosopher, regarded as the most influential logician of the 20th century Gödel’s first interest in logic was sparked when he attended a seminar on Russell’s book Introduction to Mathematical Philosophy Later he attended a lecture by David Hilbert on completeness and consistency of mathematical systems, where Hilbert posed the question of whether there is a consistent formal system of axioms of first-order logic which can derive every valid – that is, true in all models – statement of first-order logic Gödel chose this problem as the topic of his doctoral work and he completed his doctoral dissertation under the supervision of Hans Hahn in 1929, at the age of 23 In his thesis he proposed an axiomatic system for the first-order predicate logic and established its completeness This was his first famous result, known as Gödel’s Completeness Theorem In 1931 Gödel published his most important and groundbreaking work, On Formally Undecidable Propositions of “Principia Mathematica” and Related Systems, where he proved his famous incompleteness theorems Gödel’s First Incompleteness Theorem states that any axiomatic system of first-order logic which is consistent (i.e., no contradiction can be derived in it), has an effectively recognizable (recursive) set of axioms, and is expressive enough to describe the arithmetic of the natural numbers with addition and multiplication must be incomplete; that is, there are true arithmetical statements that can be stated in the language of that system but cannot be derived from its axioms In particular, this result applies to the Peano system of axioms of the arithmetic and to Zermelo–Fraenkel set theory Gödel’s Second Incompleteness Theorem states that no such consistent system can derive a statement, formalized in the language of arithmetic, claiming its consistency Gödel’s basic idea of the proof of the incompleteness theorems was conceptually simple but extremely original It involved using, for any given formal axiomatic system of the arithmetic with an effectively enumerable set of axioms, a specially developed technique of encoding of the notions of formulae, axioms, and derivations in the language of the arithmetic, to eventually construct a formula of that language that claims that it is unprovable in the given formal system Such a formula cannot be provable in the system, for that would imply its inconsistency, and therefore what it states must be true Gödel’s incompleteness theorems had a shattering effect on the attempts to find an effectively enumerable set of axioms sufficient to derive all true statements in mathematics, beginning with the work of Frege half a century earlier, and culminating in Principia Mathematica and Hilbert’s program for formalizing of the mathematics Gödel’s theorems also had a profound effect on mathematical logic and the foundations and philosophy of mathematics They are one of the greatest achievements of 20th century mathematics; they are of fundamental importance not only for the foundations of mathematics but also for computer science, as they imply that a computer can never be programmed to derive all true mathematical statements, not even all those about the arithmetic of natural numbers