What did Kurt Godel discover?

Answer 1

Kurt Godel was a mathematician who proved several very deep theorems. His best know contribution has to do with the completeness of formal systems. A formal system (in mathematics) is a set of self-evident truths (called "the axioms") along with a set of rules with which the axioms may be manipulated. The result of the proper manipulation of the rules and axioms yields the so-called "theorems". These last may, in turn, be manipulated by the same set of rules to obtain more complex theorems. The argument may be applied to any theorem and all producible theorems correspond to mathematical truths. The idea is that in a formal system no reasoning (beyond the one required to apply the rules) is needed and, therefore, the chance of subjective interpretation errors is eliminated. It was hoped that by adequately establishing a very small set of axioms and proper "production" rules one would therefore be able to produce a "model" of a non-formal system. For instance, Russell and Whitehead (two British mathematicians) claimed that such model was contained in their opus "Principia Mathematica". And, furthermore, that from it all provable truths (theorems) regarding the natural numbers (all positive integers from 0 to infinity) could be derived. However, Godel rigorously proved that Rusell's claim did not hold for all cases. In fact, he proved that there are infinitely many cases when that (the non-provability of theorems) happens if the formal system is sufficiently powerful. In other words, even though there are formal systems in which all possible truths may be mechanically found, this is so only if the purported systems are simple enough to make them uninteresting (for a mathematician at least). The theorem was called "The Incompleteness Theorem of Formal Systems". It created quite a stir in the mathematical and philosophical worlds since what Godel had just proved was equivalent to proving that "truth" may not be found (even in principle) mathematically (at least in the formal sense). Or, equivalently, that there are mathematical truths which will forever remain unknown. Since the claim was (and is) that all science may be expressed in some kind of mathematical model Godel's theorem implied that there is no way to find all truths by means of a systematic (i.e. mechanical) program. A corollary of the above is that there is no way to produce Artificial Intelligence in the sense of a machine being able to achieve a mental development similar to that of a human being. For since any machine must obey a set of physical "hard" rules (or "laws") it is forever marred by Godel's argument of incompleteness. Thus, there are truths that no machine may find that, however, may be found by a human being (such as Godel himself). Therefore, IA is unreachable even in principle. Therefore, human beings do not obey formal dictates. There are other incompleteness theorem which Godel also proved (among his many contributions). The one described, however, is the one that is best known.