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Noah Stephens-Davidowitz - Halting Problem, Incompleteness, and the Limits of Mathematics

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Uploaded on Aug 4, 2014

Courant Splash, or cSplash, is an annual one-day lecture series taught by enthusiastic students, faculty, and others associated with the Courant Institute of New York University. Since 2006, cSplash has been providing a meeting ground for mathematically inclined students in the New York metropolitan area, exposing them to the beauty, utility, and ubiquity of mathematical ideas.

Lecturer info: Noah Stephens-Davidowitz (noahsd [at] gmail [dot] com), Courant Computer Science, 2nd Year PhD Student

Anyone who's spent much time programming knows that it's surprisingly easy (and very frustrating!) to accidentally write programs that never stop running. To save ourselves some time and embarrassment, we'll ask a very simple question: Given some computer program, how can we tell whether it will run forever? Unfortunately, this basic question will prove to be maddeningly difficult to answer. We'll quickly find that computer programs can go into infinite loops in dizzyingly complex ways. Without intending to do so, we'll somehow find ourselves talking about some seemingly unrelated problems in mathematics that have stumped mathematicians for centuries. We'll actually learn that we were trying to solve them without realizing it. Oops... This will lead us, by way of Alan Turing's famous Halting Problem, to a simple proof of one of the most profound and humbling results in the philosophy of mathematics: Kurt Godel's First Incompleteness Theorem. Namely, we will show that, in a sense, NO formal system can encompass all of mathematics. We'll ponder the philosophical meaning of this statement and, if we have time, conclude by considering the statement "Nothing that is interesting can be defined unambiguously."

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