Godel's theorem

Abstract

At the beginning of the 20th century the mathematician David Hilbert posed a set of problems to the mathematical community that should have been the so-called road map oftasks to accomplish during the following hundred years. Among them was a problem which he posed in collaboration with Ackermann dealing with the question of whether a formal system of mathematical logic can be considered complete - where completeness implies that every true statement can be expressed within the system, possibly without a paradox. This was probably inspired by the recent discovery ofa series of paradoxes in Russell and Whitehead's Principia Mathematica which is now a de facto standard for defining and proving mathematical statements. The well-known Russell's paradox - formulated in a hundred different ways - has been catered for by denying the possibility of having a set being a member of itself However, other forms of paradoxes are not that easy to eliminate. Epimenides' paradox falls into this category: "I am a liar" or in logic-speak: "This statement is false". Godel's seminal work in 1931 not only managed to show that the PM system was inconsistent, but that any sufficiently powerful formal system is bound to be littered with paradoxes. It is worth stating how series this matter is: practically speaking he stated that there might exist theorems that cannot be proved or disproved - theorems about number theory itself, for instance. The approach to Godel's proofI am going to use is a simplified version based on the work of Douglas R. Hofstadter, "Godel, Escher, Bach: an Eternal Golden Braid". A book which I thoroughly recommend to anyone interested in the question of how animate matter can result out of combinations of inanimate matter.