CPT : What is the importance of Gödel's work?
CPT : What is the importance of Gödel's work?
JYB: Gödel is a key figure of modern logic. He has proven many important theorems, such as the completeness theorem for first-order logic. But he is especially famous for the incompleteness theorem(s) showing that it is not possible to axiomatize arithmetic and prove its consistency. By the way, Tarski played an important role for these theorems, discussing with Gödel in Vienna shortly before Gödel proved them (see details here).
JYB: Gödel is a key figure of modern logic. He has proven many important theorems, such as the completeness theorem for first-order logic. But he is especially famous for the incompleteness theorem(s) showing that it is not possible to axiomatize arithmetic and prove its consistency. By the way, Tarski played an important role for these theorems, discussing with Gödel in Vienna shortly before Gödel proved them (see details here).
Gödel commented his achievement as follows: “In 1678 Leibniz made a claim of the universal characteristic. In essence it does not exist, any systematic procedure for solving problems of all kinds must be nonmechanical. My incompleteness theorem makes it likely that mind is not mechanical” (see Hao Wang, 2016). Leibniz was Gödel’s favorite philosopher because he thought that Leibniz was 100% wrong. To be 100% right is nearly impossible and to be sometimes wrong, sometimes wright, is very easy. This is the general state of confusion of human mind.
Gödel commented his achievement as follows: “In 1678 Leibniz made a claim of the universal characteristic. In essence it does not exist, any systematic procedure for solving problems of all kinds must be nonmechanical. My incompleteness theorem makes it likely that mind is not mechanical” (see Hao Wang, 2016). Leibniz was Gödel’s favorite philosopher because he thought that Leibniz was 100% wrong. To be 100% right is nearly impossible and to be sometimes wrong, sometimes wright, is very easy. This is the general state of confusion of human mind.
Gödel’s first incompleteness theorem is often presented as a negative result, but it is indeed good news: human intelligence will never be replaced by the artificial intelligence of a machine. Computers are certainly very useful, but they cannot substitute human beings for reasoning.
Gödel’s first incompleteness theorem is often presented as a negative result, but it is indeed good news: human intelligence will never be replaced by the artificial intelligence of a machine. Computers are certainly very useful, but they cannot substitute human beings for reasoning.