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PROFESSOR: [? Diagonal ?] arguments are elegant, and infinite sets-- some people think-- are romantic. But you could legitimately ask what is all this weird infinite stuff doing in a course that's math for computer science? And the reason is that diagonal arguments turn out to play a fundamental role in the theory of computing. And what we're going to talk about now is the application of diagonal arguments to show that there are noncomputable sets and examine a particular one. So let's look at the class of infinite binary strings. Now, we've seen that there are an uncountable number of infinite binary strings, and that's because there was a simple bijection between the infinite binary strings and the subsets of the natural numbers-- that is the power set of n. Let's look at the infinite binary strings that we might think of and call computable strings. And what I mean by a computable string is that there's simply a procedure that will tell me w...