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UnpThms110509 - UNPROVABLE THEOREMS by Harvey M Friedman...

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UNPROVABLE THEOREMS by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University [email protected] http://www.math.ohio-state.edu/ ~friedman/ Boston Mathematics Colloquium delivered at MIT October 8, 2009
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It’s great to be back here! I was a math student 1964-67, and remember taking courses here in 2-190, and in 2-290! I started as a Freshman and finished with a Ph.D. MIT decided to get rid of me early! I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290? The textbook was Elliot Mendelson’s, Introduction to Mathematical Logic, still a good textbook today. I knew that logic was supposed to be the basis of all thinking (maybe a bit naive considering, e.g., the political world). I remember asking Hilary Putnam - then a Professor in the Philosophy Department here - “how does logic begin?” Our meeting was outside Walker Memorial Cafeteria. I’m still pondering that one. And I remember talking about recursion theory inside my thesis advisor’s fancy new sports car. The proud owner was Gerald Sacks, then at MIT.
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What This Is About: The Search When I was a student way back in 1964, I was fascinated by the drama created by the great legendary figure Kurt Gödel (died 1978): there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics. Furthermore, Gödel showed that this cannot be repaired, in the following sense: even if we add finitely many new axioms to the usual axioms and rules of inference of mathematics, there will remain mathematical statements that cannot be proved or refuted. These startling results are taught in the usual mathematical logic curriculum. One common way of proving these results provides no examples. So what about the examples? I.e., examples of such INCOMPLETENESS ?
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STANDARD EXAMPLES OF INCOMPLETENESS 1. That “the usual axioms and rules of inference for mathematics does not lead to a contradiction”. I.e., “ZFC does not have a contradiction” is neither provable nor refutable in ZFC. 2. That “every infinite set of real numbers is either in one-one correspondence with the integers or in one-one correspondence with the real line”. I.e., “the continuum hypothesis of Cantor” is neither provable nor refutable in ZFC. These and related examples appear in the mathematical logic curriculum. Note that these examples are very much associated with abstract set theory, and unusually far removed in spirit and content from traditional down to earth mathematics. I was very aware of this disparity, even as a student, which was reinforced in conversations with other students and Professors.
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