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Unformatted text preview: CONCRETE MATHEMATICAL INCOMPLETENESS by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University [email protected] University of Cambridge Cambridge, England November 8, 2010 minor revision: November 18, 2010 What This Is About: The Search When I was a student (long time ago), 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 ? 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 oneone correspondence with the integers or in oneone 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. For several decades I have been seeking examples of a new “down to earth” kind. This has been an ongoing process. Recently, there has been some particularly clear progress. I will highlight the main events up through now. WHAT IS AN UNPROVABLE THEOREM? All of the examples of Concrete Incompleteness that we are going to talk about, come under the category of what we call UNPROVABLE THEOREMS . An Unprovable Theorem is a theorem that is i. proved using a by now well studied hierarchy of additional axioms for mathematics called the “large cardinal hierarchy”. ii. cannot be proved (or refuted) with only the usual axioms for mathematics. A highlight of this talk is the presentation of some examples of Unprovable Theorems of a radically new kind. These will take the form of structural properties of kernels in digraphs. DOES THIS TALK HAVE ANYTHING TO DO WITH THE AXIOM OF CHOICE?...
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 Fall '08
 JOSHUA
 Math, Set Theory, Inaccessible cardinal, ZFC, boolean relation theory

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