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Aaron Goldsmith 1 Gödel's Incompleteness Theorems David Hilbert put considerable effort into finding a complete set of axioms for all of mathematics. In fact, this was Hilbert's second problem, one of the ten presented at the International Congress in 1900. Such a set would unify mathematics and place everyone on the same page as to what is assumed to be true without proof. A logician named Kurt Gödel, however, foiled Hilbert's program with two "incompleteness theorems" in 1931. These theorems say that any formal system that is advanced enough to prove Peano arithmetic is either inconsistent or incomplete. That is, either you arrive at a contradiction, or there exists a true statement which cannot be proved in the system. Alfred North Whitehead and Bertrand Russell set out in "The Principia Mathematica" to complete the foundation of mathematics, when around the same time, Gödel proved that they were wasting their time. Sadly, when Gödel's paper was first published, it made no sense to even most mathematicians, but Whitehead and Russell's was not so esoteric. The logic and reasoning in Gödel's proof at the time was so novel that only those specialized with the highly technical literature could understand the full implications [3]. Basically, Gödel used half his proof to show that the four axioms for second-order arithmetic were incomplete (there exists a true statement that cannot be proven), and from there set a correlation between Peano arithmetic and the symbols of the language. Gödel lined up all the symbols, operators, axioms, and everything about the system, then assigns positive integers to each one. This means any formal proof can be represented with a string of integers [1]. The other half of the proof involves finding a statement that could not be proved or disproved. It turns out that he used a self referential statement similar to, "I am lying," though this is not directly feasible in a formal proof. In this way, he showed that no finite set of axioms was Aaron Goldsmith 2 sufficient for proving all true statements in any system that invokes Peano arithmetic. Ironically, the very cornerstone of Gödel's proof looks similar to Russell's Paradox, from the same Russell that was pursuing the opposite result as Gödel. Russell's Paradox is like a town with a barber that shaves precisely those that do not shave themselves. The barber must and cannot shave himself, a more problematic situation than when we can't decide on which pizza place to support. This inconsistency theorem is sometimes used to say the Hebrew Bible as well as any other is either incomplete or inconsistent (contradictory), by regarding the Bible as a finite set of axioms. This is a completely unnecessary argument. Gödel's theorems are from formal logic and should only be used in formal mathematical systems. The Bible does not claim to decide whether irrelevant statements are true or not. Thus, incompleteness is not as applicable as many think. For example, if I asked the Bible to decide whether or not the statement "Peter had a large mole on his left cheek" is true, it would give me nothing. That is not what the Bible concerns. The same argument is used for the United States Constitution, the English language, and many other documents [4]. And, the same rebuttal applies. There is also much debate over whether Gödel's first inconsistency theorem means that artificial intelligence will never reach the level of the human brain [3]. This assumes that the human brain is capable of knowing truths without proof and that artificial intelligence uses sufficiently advanced mechanisms to invoke Gödel's Theorem. Clearly, Gödel rocked the mathematical world with his discoveries. When he was awarded an honorary degree from Harvard University in 1952, his work was described as "one of the most important advances in logic in modern times" [3]. Aaron Goldsmith 3 Bibliography [1] David Hofstadter. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, New York, 20th Anniversary Ed, 1999. [2] Raymond Smullyan. Gödel's Incompleteness Theorems. Oxford UP, New York, 1992. [3] Ernest Nagel and James R. Newman. Gödel's Proof. New York UP, New York, 1958. pp 3, 100. [4] http://en.wikipedia.org/wiki/Incompleteness_theorem#Discussion_and_implications