In Intermediate Analysis this morning, three students each went to the board to present homework problems on basics of sets and mappings. This is first time that I have incorporated the oral presentation of homework as a part of an upper level mathematics course. It was one of my least favorite aspects of the graduate courses that I took and yet I realize just how formative it was in the development of my understanding for those courses. I have a stronger grasp of those courses in which such presentation was an essential portion of the course.
The lecture for today covered the concepts of denumerable (countably infinite) and countable sets. We proved that some basic sets are denumerable, such as the natural numbers, even numbers, odd numbers, the cartesian product of countable sets, . . . We did not quite get to the proof that the reals are uncountable but that should happen at the beginning of the next class. I've decided to add some additional information to the course that is not included in the text at this point. We are going to discuss cardinal numbers and their ordering. We'll broach the subject of the Continuum Hypothesis in the next class.
I can recall the first time I encountered the idea that the concepts of infinity with respect to infinite limits and limits at infinity are distinct from the cardinality, or "size", of infinite sets. It blew my mind when I first learned that the sets and are, in fact, the same size, even though only half the first set appears in the second set. But they are the same size of "infinity" while the number of real numbers is larger. That is, it is a strictly larger infinity than the number of counting numbers. All of this good stuff comes next time in class.
By the way, I have to get better about not ending class in the middle of a proof. A more judicious use of my time is needed. (Like that will ever happen). I am already doing poor on my dismissal of class on time. Although, the latest I've let them out is 3 minutes late. I will work on that as well.