[FOUNTAIN] Incompleteness, a Job Well Done

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[FOUNTAIN] Incompleteness, a Job Well Done

"Magic King of Mathematics", "God of Mathematics on Earth" are the nicknames of Czechoslovakia-born Kurt Goedel, one of the most influential mathematicians and logicians of the 20th century. Goedel was born in 1906 and died in 1978, and last Saturday was the anniversary of his birth.

In 1930, Goedel, at age 24, announced the "Incompleteness Theorem," shocking his colleagues and contemporary thought on a range of subjects.

Goedel demonstrated that within any given branch of mathematics, there always will be some propositions that could not be proven either true or false using the rules and axioms of that mathematical branch itself (the first incompleteness theory). Hence, one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions. (the second incompleteness theory) Goedel proved this with strict mathematical methods.

The main goal of leading mathematicians in the twentieth century was to prove that there is no contradiction in mathematics. This task stemmed from Georg Cantor's creation of the Set Theory at the end of the 19th century. When Cantor was examining the characteristics of infinitive sets, he came up with the paradox that the size of the whole equals the size of its parts. Since the Set Theory covered all aspects of mathematics, the existence of paradox can shatter the roots of mathematics. This gave birth to mathematical basic theory, which studies the principles and methods of mathematics. Despite the paradox of the Set Theory, it was widely believed that mathematics was complete and that all mathematical hypotheses could be proven in the system.

But, Goedel's Incompleteness Theorem broke down this belief. If the most complete structure and knowledge of mathematics can be broken, what would happen to science and other knowledge?

Incompleteness Theorem and Heisenberg's Uncertainty Principle, which says it is theoretically impossible to accurately estimate the position and energy of protons, are the two theories that proved the limitation of human knowledge. Goedel proved the completeness theory, which says propositions that are logically correct can be proven in the logical system. Goedel also proved the absence of contradiction in the continuity hypothesis about the density of the set. Kurt Goedel emigrated from Austria to the United States in 1940 and was naturalized in 1948. He became a member of the Institute for Advanced Study at Princeton University in New Jersey. Goedel served as a professor at the Institute until 1953.

The world's loss is not only Goedel and his brother Rudolf Goedel, a radiologist. They did not have offspring. If it had been possible to freeze their sperm, people would be lining up for the sperm now.

by Cho Hyon-wook

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